The Wolfram Alpha Logo

 I used Wolfram Alpha today to determine when I would be 77.77777 years old. 

Yep, 9:21pm on January 11th 2027, midway between my father's birthday on January 9th and my brother Tony's birthday on January 13th. More about that later but, perhaps because of my recent foray into sketching, I happened to notice the logo shown below that appears on the web page. Oddly, I failed to notice the red stellated polyhedra that appears in the image above, maybe because it's quite small.

I asked Gemini the following:

What can you tell me about the Wolfram Alpha logo. It’s an interesting shape. Does it have any history or special significance? Why was this shape shown? Does it have name?

This was the response:

The shape shown is a 2D stylized projection of a \( \textbf{rhombic hexecontahedron}\).

Mathematical History and Properties

• Origins: The shape was formally described mathematically in 1940 by the German mathematician Helmut Unkelbach.

• Geometric Structure: It is a \( \textbf{non-convex zonohedron} \) composed entirely of 60 intersecting \( \textbf{golden rhombi} \) (rhombuses whose diagonals align exactly with the golden ratio).

• Vertices and Edges: The structure features 62 vertices and 120 edges, all organized with perfect \( \textbf{icosahedral symmetry} \).

• Stellation: It is formed as a stellation of the \( \textbf{rhombic triacontahedron} \), meaning it is created by extending the face planes of a core polyhedron until they intersect again to form a new, more complex outer boundary.

• Internal Geometry: The internal volume of the shape can be cleanly dissected into 20 acute \( \textbf{golden rhombohedra} \) that meet exactly at the center point.

Significance and Selection as a Logo

• The "Spikey" Tradition: The rhombic hexecontahedron continues a decades-long corporate tradition initiated by Stephen Wolfram. Since the release of Mathematica 1.0 in 1988, Wolfram products have been represented by complex, mathematically generated 3D polyhedra, affectionately known internally as "Spikeys." While early Mathematica logos featured shapes like the stellated icosahedron and hyperbolic dodecahedron, the rhombic hexecontahedron was designated specifically for the launch of the Wolfram|Alpha engine in 2009.

• Visualizing Computation: The logo was selected to reflect the computational complexity, algorithmic precision, and mathematical rigor underpinning Wolfram|Alpha. The shape’s composition—made of identical, mathematically strict intersecting faces that combine to form a highly intricate and globally symmetric structure—serves as a visual metaphor for the engine's ability to process massive amounts of disparate raw data and compute unified, structured answers.

Here is what Gemini did with a sketch that I made of the shape using the following prompt:

This image is a 2D stylized projection of a rhombic hexecontahedron, used by Wolfram Alpha as its logo. I need you impose a strict symmetry on the shape and make it look as solid and 3D-looking as possible with shading or whatever. The faces I’ve represented as windows and these should be rendered as translucent with a yellow light emanating, seemingly from within. The overall shape should then look like a polyhedral house with a warm and inviting glow.

Instagram Link

The Golden Rhombus and Golden Rhombohedron referred to earlier look this:

A rhombohedron shown above is an acute golden rhombohedron and it is a cube that has been pushed out of shape so that all of its faces are rhombi. A golden rhombus has a ratio of long diagonal to short diagonal of \( \phi \; : \; 1\). A golden rhombohedron is simply a rhombus with all faces golden rhombi. 

• The acute angle in a golden rhombus is given by \( 2 \text{arctan} (1/ \phi) \approx 63.43^{\circ} \)
• The obtuse angle is given by \( 2 \text{arctan} ( \phi) \approx 116.57^{\circ} \) 

Twenty acute golden rhombohedra can be combined to form a solid rhombic hexecontahedron. There is also an obtuse golden rhombohedron but I'll deal with that later.

28215: An Interesting Number

I've heard it said that all numbers are interesting and that, if a number is not, then it's interesting because it's not. The number associated with my diurnal age today (\( \textbf{28215}\) ) is definitely interesting. 

FIRST INTERESTING PROPERTY

Its prime factorisation is as follows:$$28215=3^3 \times 5 \times 11 \times 19$$It has 32 proper divisors and these are:

1, 3, 5, 9, 11, 15, 19, 27, 33, 45, 55, 57, 95, 99, 135, 165, 171, 209, 285, 297, 495, 513, 627, 855, 1045, 1485, 1881, 2565, 3135, 5643, 9405

The sum of these divisors is 29385 and so the number is abundant because this sum exceeds the number itself. Furthermore, all of these divisors are deficient and this makes it primitive abundant. Lastly the number is odd. This makes 28215 an odd primitive abundant number and these sorts of numbers are quite rare. Here is the list of the 50 such numbers up to 40000 (permalink):

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, \( \textbf{28215}\), 29835, 30555, 31395, 31815, 32445, 33345, 33495, 33915, 34155, 35805, 39585

SECOND INTERESTING PROPERTY

It can be seen from its factorisation that 28215 is 6-almost prime and its reversal, 51282, is also 6-almost prime:$$ \begin{align} 28215 &=3^3 \times 5 \times 11 \times 19 \\ 51282 &= 2 \times 3^2 \times 7 \times 11 \times 37 \end{align}$$While this property is not quite are rare as being odd primitive abundant, there are still only 118 such numbers in the range up to 40000. These are (permalink):

2576, 2772, 2970, 2992, 4284, 4356, 4410, 4600, 4698, 4824, 5265, 5625, 6534, 6752, 6776, 6900, 8008, 8250, 8964, 10710, 10890, 13140, 13986, 16236, 16335, 17577, 18504, 19494, 20286, 20574, 21112, 21114, 21150, 21160, 21336, 21492, 21576, 21609, 21712, 21900, 21912, 21996, 22392, 22770, 22788, 22824, 22869, 23058, 23247, 23250, 23496, 23562, 23580, 23598, 23632, 23832, 24156, 24660, 24975, 25020, 25092, 25104, 25164, 25245, 25300, 25416, 25434, 25452, 25608, 25668, 25752, 25952, 26163, 26334, 26532, 27060, 27108, 27135, 27192, 27240, 27248, 27270, 27405, 27408, 27468, 27472, 27588, 27608, 27636, 27816, 28116, \( \textbf{28215} \), 28314, 28710, 28782, 28890, 29052, 29172, 29322, 29340, 29392, 29412, 29580, 29750, 29784, 29835, 29900, 29960, 29984, 32967, 34965, 35775, 35937, 36162, 36990, 37026, 38367, 38934

I explore this reversibility extensively in my post titled Beyond Emirp from July 2025.

THIRD INTERESTING PROPERTY

The totient of a number \(n\) (commonly known as Euler's totient function or phi function, denoted as \(\phi(n)\)) counts the number of positive integers up to \(n\) that share no common factors with \(n\) other than 1. These numbers are called "relatively prime" or "coprime" to \(n\).

28215 has the property that it and its reversal, 51282, both have the same totient. Thus:$$ \phi(28215)=\phi(51282)=12960 $$This property is the rarest of all because in the range up to 40000 there are only 25 numbers with this property. They are:

190, 427, 429, 724, 924, 4147, 4697, 6276, 6726, 7414, 7964, 9079, 9709, 10040, 10940, 14450, 15860, 19190, 20493, 20553, 28092, \( \textbf{28215}\), 29082, 35502, 39402

FOURTH INTERESTING PROPERTY

28215 is a member of OEIS  A076773:

A076773   2-nadirs of \( \phi\) : numbers \( k \text{ such that }\)  

\( \phi(k-2) \gt \phi(k-1) \gt \phi(k) \lt \phi(k+1) \lt \phi(k+2) \)

I covered this in a post titled Totient Function: Jagged Versus Rounded Local Minima back in March of 2025. Figure 1 gives an idea of what is going on:

Figure 1

Here the \(\phi\) values for 28213, 28214, 26215, 28216, 28217 are:$$27808 \lt14106 \lt 12960 \lt14104 \lt23184$$There are 238 such minima in the range up to 40000:

315, 525, 735, 1155, 1365, 1575, 1755, 1785, 1815, 1995, 2145, 2415, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4305, 4515, 4725, 4935, 5115, 5145, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7095, 7245, 7395, 7455, 7605, 7665, 8085, 8265, 8505, 8715, 8745, 8925, 9135, 9345, 9405, 9555, 9735, 9765, 9975, 10185, 10395, 10455, 10545, 10815, 10965, 11055, 11235, 11385, 11445, 11655, 11865, 12075, 12285, 12495, 12675, 12705, 12915, 13125, 13335, 13545, 13695, 13965, 14025, 14175, 14355, 14385, 14595, 14805, 14835, 15015, 15045, 15225, 15405, 15435, 15645, 15675, 15855, 16005, 16065, 16275, 16335, 16485, 16695, 16905, 17085, 17325, 17355, 17745, 17955, 18135, 18165, 18375, 18585, 18645, 18795, 18975, 19215, 19425, 19635, 19665, 20055, 20265, 20295, 20475, 20625, 20685, 20865, 20895, 21105, 21255, 21315, 21525, 21945, 22365, 22425, 22575, 22605, 22785, 22995, 23205, 23265, 23415, 23595, 23625, 23655, 23835, 23985, 24225, 24255, 24675, 24885, 24915, 25095, 25245, 25305, 25515, 25575, 25725, 25905, 25935, 26145, 26325, 26565, 26775, 26985, 27027, 27195, 27615, 27825, 27885, 28035, \( \textbf{28215}\), 28245, 28275, 28455, 28665, 28815, 28875, 29055, 29295, 29505, 29865, 29925, 30195, 30345, 30555, 30723, 30765, 30975, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32655, 32835, 32895, 33033, 33075, 33345, 33495, 33705, 33735, 33915, 34125, 34155, 34335, 34485, 34515, 34545, 34755, 34965, 35175, 35385, 35805, 36225, 36435, 36465, 36645, 36795, 36855, 37065, 37275, 37455, 37485, 37695, 37905, 38115, 38535, 38745, 38775, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

FIFTH INTERESTING PROPERTY

28215 is also a member of A323380:

A323380  2-zeniths of \(\sigma\): numbers \( k \text{ such that }\)

                    \( \sigma(k-2) \lt \sigma(k-1) \lt \sigma(k) \gt \sigma(k+1) \gt \sigma(k+2) \)

I covered this in a post titled Totient and Sigma Graphs Revisited in August of 2025. Figure 2 shows what's going on and it's the local zenith replacing the local nadir of the totient graph:

Figure 2

Here the sigma values for 28213, 28214, 26215, 28216, 28217 are:$$28620 \lt 42324 \lt 57600 \gt52920 \gt 33600$$Below is a list of numbers up to 40000 that belong to BOTH OEIS A323380 and OEIS A076773 (sigma and totient respectively):

315, 525, 1155, 1575, 1755, 1785, 1995, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4515, 4725, 5115, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7245, 7605, 8085, 8505, 8715, 8925, 9135, 9405, 9555, 9765, 9975, 10395, 11235, 11385, 11445, 11655, 12075, 12285, 12675, 12705, 12915, 13125, 13545, 13965, 14025, 14175, 14355, 14595, 14805, 15015, 15435, 15645, 15675, 16005, 16065, 16275, 16335, 16695, 16905, 17325, 17745, 17955, 18135, 18375, 18585, 18795, 19215, 19635, 20475, 20685, 21105, 21315, 21525, 21945, 22365, 22605, 22995, 23205, 23595, 23625, 23835, 24255, 24675, 24885, 24915, 25245, 25515, 25725, 25935, 26325, 26565, 26775, 27027, 27195, 27885, 28035, \( \textbf{28215}\), 28245, 28275, 28665, 28875, 29295, 29925, 30195, 30345, 30555, 30723, 30765, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32835, 33075, 33345, 33495, 33915, 34125, 34155, 34485, 34515, 34755, 34965, 35175, 35805, 36225, 36435, 36645, 36795, 36855, 37275, 37485, 38115, 38745, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

SIXTH INTERESTING PROPERTY

28215 has a totient and sum of divisors that have 2, 3 and 5 as their distinct prime factors:$$ \begin{align} \sigma(28215) &= 57600 = 2^8 \times 3^2 \times 5^2 \rightarrow 2, 3, 5 \text{ as distinct prime factors} \\ \phi(28215) &= 12960 = 2^5 \times 3^4 \times 5 \rightarrow 2, 3, 5 \text{ as distinct prime factors} \end{align} $$There are 143 such numbers in the range from 28215 to 40000:

\( \textbf{28215} \), 28258, 28329, 28340, 28424, 28458, 28614, 28728, 28768, 28782, 28809, 28826, 28985, 29029, 29222, 29260, 29295, 29337, 29393, 29512, 29640, 29667, 29678, 29835, 29848, 30039, 30184, 30240, 30264, 30305, 30381, 30504, 30566, 30760, 30780, 30814, 30888, 30914, 30943, 30956, 30996, 31008, 31027, 31160, 31174, 31283, 31331, 31392, 31416, 31465, 31496, 31529, 31806, 31816, 32103, 32130, 32131, 32298, 32376, 32395, 32589, 32604, 32718, 32802, 32984, 33015, 33176, 33292, 33345, 33383, 33440, 33480, 33495, 33497, 33528, 33572, 33592, 33836, 33885, 33915, 34008, 34162, 34276, 34293, 34317, 34440, 34452, 34573, 34580, 34605, 34782, 34884, 35061, 35074, 35112, 35340, 35343, 35424, 35464, 35530, 35752, 35805, 35910, 35948, 35960, 36366, 36423, 36666, 36828, 36859, 36860, 36890, 36920, 37060, 37128, 37417, 37638, 37719, 37730, 37758, 37772, 37961, 38038, 38152, 38285, 38340, 38368, 38408, 38610, 38745, 38760, 38874, 39032, 39121, 39219, 39270, 39370, 39458, 39501, 39520, 39556, 39576, 39729

SEVENTH INTERESTING PROPERTY

28215 is what I've termed an \(a,b,c,d\) number because it can be combined with three other numbers, all with the same digits, to form a simple additive equation and this can be done in two different ways. Here is what I mean:$$ \begin{align} 25182 + 28125 + \textbf{28215} &= 81522 \\ 25812 + 28125 + \textbf{28215} &= 82152 \end{align}$$Here are the numbers with this property in the range from 28215 to 40000:

\( \textbf{28215}\), 28260, 28269, 28359, 28413, 28458, 28467, 28476, 28512, 28521, 28539, 28548, 28593, 28611, 28647, 28674, 28692, 28701, 28710, 28719, 28746, 28764, 28791, 28845, 28854, 28863, 28917, 28935, 28953, 28962, 28971, 29016, 29034, 29043, 29061, 29106, 29160, 29178, 29187, 29268, 29304, 29340, 29358, 29367, 29385, 29394, 29439, 29448, 29475, 29493, 29538, 29583, 29601, 29610, 29628, 29637, 29673, 29682, 29718, 29754, 29763, 29781, 29817, 29835, 29853, 29871, 29961, 30015, 30150, 30159, 30168, 30195, 30285, 30294, 30429, 30492, 30519, 30582, 30591, 30627, 30681, 30726, 30825, 30852, 30924, 30942, 30951, 31059, 31068, 31149, 31158, 31176, 31185, 31464, 31491, 31509, 31590, 31599, 31608, 31635, 31644, 31653, 31680, 31689, 31698, 31761, 31788, 31806, 31815, 31842, 31860, 31869, 31878, 31896, 31905, 31959, 31968, 31986, 31995, 32049, 32076, 32085, 32148, 32418, 32481, 32490, 32499, 32580, 32607, 32679, 32697, 32760, 32769, 32796, 32814, 32841, 32850, 32859, 32886, 32895, 32904, 32958, 32967, 32976, 32985, 32994, 34029, 34119, 34128, 34164, 34182, 34218, 34281, 34299, 34461, 34614, 34641, 34812, 34821, 34911, 34992, 35019, 35082, 35091, 35109, 35118, 35190, 35217, 35271, 35631, 35712, 35721, 35802, 35820, 35829, 35892, 35910, 35982, 35991, 36018, 36108, 36117, 36135, 36144, 36153, 36171, 36198, 36261, 36279, 36288, 36297, 36315, 36351, 36414, 36513, 36531, 36621, 36711, 36729, 36792, 36810, 36819, 36918, 36927, 36972, 36981, 37116, 37125, 37161, 37179, 37197, 37215, 37251, 37269, 37296, 37521, 37611, 37629, 37719, 37917, 37962, 38016, 38061, 38106, 38115, 38142, 38151, 38160, 38169, 38187, 38196, 38214, 38241, 38286, 38295, 38412, 38511, 38529, 38592, 38610, 38619, 38682, 38691, 38817, 38826, 38871, 38916, 38925, 38952, 38961, 39015, 39024, 39042, 39051, 39105, 39150, 39159, 39177, 39186, 39195, 39204, 39258, 39285, 39402, 39411, 39420, 39492, 39501, 39510, 39528, 39582, 39591, 39618, 39627, 39672, 39681, 39717, 39726, 39762, 39816, 39825, 39852, 39861, 39942, 39951

EIGHTH INTERESTING PROPERTY

If the prime factors, with multiplicity, of 28215 are concatenated in ascending order, they form a prime number. Thus:$$28215 =3^3 \times 5 \times 11 \times 19 \rightarrow 33351119$$Such a prime is called the home prime and so 28215 is only one step removed from its home prime. There are many other concatenations that yield primes and all the possibilities are listed below (permalink):

11193353, 11319533, 11335193, 19331153, 19335311, 19351133, 19511333, 31119353, 31131953, 31133519, 31153193, 31933511, 31951133, 31953113, 33191153, 33195311, 33311519, \( \textbf{33351119} \), 35113193, 35191133, 35193311, 35319113, 35331119, 35331911, 51131933, 51133193

This property, of being one step removed from its home prime, is relatively common but nonetheless interesting.

NINTH INTERESTING PROPERTY

28215 is a member of the commas sequence beginning with 8. I explore sequences of this type in my blog post The Commas Sequence from December of 2023. The full trajectory up to 40000 is as follows:

[8, 97, 168, 250, 252, 274, 317, 390, 393, 427, 502, 527, 603, 639, 736, 804, 852, 880, 888, 977, 1048, 1129, 1220, 1221, 1232, 1253, 1284, 1325, 1376, 1437, 1508, 1589, 1680, 1681, 1692, 1713, 1744, 1785, 1836, 1897, 1968, 2050, 2052, 2074, 2116, 2178, 2260, 2262, 2284, 2326, 2388, 2470, 2472, 2494, 2536, 2598, 2680, 2682, 2704, 2746, 2808, 2890, 2892, 2914, 2956, 3019, 3112, 3135, 3188, 3271, 3284, 3327, 3400, 3403, 3436, 3499, 3592, 3615, 3668, 3751, 3764, 3807, 3880, 3883, 3916, 3979, 4073, 4107, 4181, 4195, 4249, 4343, 4377, 4451, 4465, 4519, 4613, 4647, 4721, 4735, 4789, 4883, 4917, 4991, 5006, 5071, 5086, 5151, 5166, 5231, 5246, 5311, 5326, 5391, 5406, 5471, 5486, 5551, 5566, 5631, 5646, 5711, 5726, 5791, 5806, 5871, 5886, 5951, 5966, 6032, 6058, 6144, 6190, 6196, 6262, 6288, 6374, 6420, 6426, 6492, 6518, 6604, 6650, 6656, 6722, 6748, 6834, 6880, 6886, 6952, 6978, 7065, 7122, 7149, 7246, 7313, 7350, 7357, 7434, 7481, 7498, 7585, 7642, 7669, 7766, 7833, 7870, 7877, 7954, 8002, 8030, 8038, 8126, 8194, 8242, 8270, 8278, 8366, 8434, 8482, 8510, 8518, 8606, 8674, 8722, 8750, 8758, 8846, 8914, 8962, 8990, 8998, 9087, 9166, 9235, 9294, 9343, 9382, 9411, 9430, 9439, 9538, 9627, 9706, 9775, 9834, 9883, 9922, 9951, 9970, 9979, 10070, 10071, 10082, 10103, 10134, 10175, 10226, 10287, 10358, 10439, 10530, 10531, 10542, 10563, 10594, 10635, 10686, 10747, 10818, 10899, 10990, 10991, 11002, 11023, 11054, 11095, 11146, 11207, 11278, 11359, 11450, 11451, 11462, 11483, 11514, 11555, 11606, 11667, 11738, 11819, 11910, 11911, 11922, 11943, 11974, 12015, 12066, 12127, 12198, 12279, 12370, 12371, 12382, 12403, 12434, 12475, 12526, 12587, 12658, 12739, 12830, 12831, 12842, 12863, 12894, 12935, 12986, 13047, 13118, 13199, 13290, 13291, 13302, 13323, 13354, 13395, 13446, 13507, 13578, 13659, 13750, 13751, 13762, 13783, 13814, 13855, 13906, 13967, 14038, 14119, 14210, 14211, 14222, 14243, 14274, 14315, 14366, 14427, 14498, 14579, 14670, 14671, 14682, 14703, 14734, 14775, 14826, 14887, 14958, 15039, 15130, 15131, 15142, 15163, 15194, 15235, 15286, 15347, 15418, 15499, 15590, 15591, 15602, 15623, 15654, 15695, 15746, 15807, 15878, 15959, 16050, 16051, 16062, 16083, 16114, 16155, 16206, 16267, 16338, 16419, 16510, 16511, 16522, 16543, 16574, 16615, 16666, 16727, 16798, 16879, 16970, 16971, 16982, 17003, 17034, 17075, 17126, 17187, 17258, 17339, 17430, 17431, 17442, 17463, 17494, 17535, 17586, 17647, 17718, 17799, 17890, 17891, 17902, 17923, 17954, 17995, 18046, 18107, 18178, 18259, 18350, 18351, 18362, 18383, 18414, 18455, 18506, 18567, 18638, 18719, 18810, 18811, 18822, 18843, 18874, 18915, 18966, 19027, 19098, 19179, 19270, 19271, 19282, 19303, 19334, 19375, 19426, 19487, 19558, 19639, 19730, 19731, 19742, 19763, 19794, 19835, 19886, 19947, 20019, 20111, 20123, 20155, 20207, 20279, 20371, 20383, 20415, 20467, 20539, 20631, 20643, 20675, 20727, 20799, 20891, 20903, 20935, 20987, 21059, 21151, 21163, 21195, 21247, 21319, 21411, 21423, 21455, 21507, 21579, 21671, 21683, 21715, 21767, 21839, 21931, 21943, 21975, 22027, 22099, 22191, 22203, 22235, 22287, 22359, 22451, 22463, 22495, 22547, 22619, 22711, 22723, 22755, 22807, 22879, 22971, 22983, 23015, 23067, 23139, 23231, 23243, 23275, 23327, 23399, 23491, 23503, 23535, 23587, 23659, 23751, 23763, 23795, 23847, 23919, 24011, 24023, 24055, 24107, 24179, 24271, 24283, 24315, 24367, 24439, 24531, 24543, 24575, 24627, 24699, 24791, 24803, 24835, 24887, 24959, 25051, 25063, 25095, 25147, 25219, 25311, 25323, 25355, 25407, 25479, 25571, 25583, 25615, 25667, 25739, 25831, 25843, 25875, 25927, 25999, 26091, 26103, 26135, 26187, 26259, 26351, 26363, 26395, 26447, 26519, 26611, 26623, 26655, 26707, 26779, 26871, 26883, 26915, 26967, 27039, 27131, 27143, 27175, 27227, 27299, 27391, 27403, 27435, 27487, 27559, 27651, 27663, 27695, 27747, 27819, 27911, 27923, 27955, 28007, 28079, 28171, 28183, \( \textbf{28215} \), 28267, 28339, 28431, 28443, 28475, 28527, 28599, 28691, 28703, 28735, 28787, 28859, 28951, 28963, 28995, 29047, 29119, 29211, 29223, 29255, 29307, 29379, 29471, 29483, 29515, 29567, 29639, 29731, 29743, 29775, 29827, 29899, 29991, 30004, 30047, 30120, 30123, 30156, 30219, 30312, 30335, 30388, 30471, 30484, 30527, 30600, 30603, 30636, 30699, 30792, 30815, 30868, 30951, 30964, 31007, 31080, 31083, 31116, 31179, 31272, 31295, 31348, 31431, 31444, 31487, 31560, 31563, 31596, 31659, 31752, 31775, 31828, 31911, 31924, 31967, 32040, 32043, 32076, 32139, 32232, 32255, 32308, 32391, 32404, 32447, 32520, 32523, 32556, 32619, 32712, 32735, 32788, 32871, 32884, 32927, 33000, 33003, 33036, 33099, 33192, 33215, 33268, 33351, 33364, 33407, 33480, 33483, 33516, 33579, 33672, 33695, 33748, 33831, 33844, 33887, 33960, 33963, 33996, 34059, 34152, 34175, 34228, 34311, 34324, 34367, 34440, 34443, 34476, 34539, 34632, 34655, 34708, 34791, 34804, 34847, 34920, 34923, 34956, 35019, 35112, 35135, 35188, 35271, 35284, 35327, 35400, 35403, 35436, 35499, 35592, 35615, 35668, 35751, 35764, 35807, 35880, 35883, 35916, 35979, 36072, 36095, 36148, 36231, 36244, 36287, 36360, 36363, 36396, 36459, 36552, 36575, 36628, 36711, 36724, 36767, 36840, 36843, 36876, 36939, 37032, 37055, 37108, 37191, 37204, 37247, 37320, 37323, 37356, 37419, 37512, 37535, 37588, 37671, 37684, 37727, 37800, 37803, 37836, 37899, 37992, 38015, 38068, 38151, 38164, 38207, 38280, 38283, 38316, 38379, 38472, 38495, 38548

So, overall, 28215 is a very interesting number.

Horadam Sequences

A Horadam sequence is a generalization of the Fibonacci numbers defined by the four constants (\(p,q,r,s\)) and the definitions \(H_0=p\) and \(H_1=q\) together with the linear recurrence equation for \(n>1\):$$H_n=sH_{n-1}+rH_{n-2}$$Specific values of \(r\) and \(s\) lead to well known sequences:

• Fibonacci Numbers: \(F_n=F_{n-1}+F_{n-2}\\ \text{ where }n \geq 2, F_0=0, F_1=1\)
   
• Lucus Number: \(L_n=L_{n-1}+L_{n-2} \\ \text{ where } n \geq 2, L_0=2, L_1=1\)
  
  
• Pell Numbers: \(P_n=2P_{n-1}+P_{n-2} \\ \text{ where } n \geq 2, P_0=0, P_1=1\)
  
  
• Pell-Lucus Numbers: \(Q_n=2Q_{n-1}+Q_{n-2} \\ \text{ where } n \geq 2, Q_0=Q_1=1\)
  
  
• Jacobsthal Numbers: \(J_n=J_{n-1}+2J_{n-2} \\ \text{ where } n \geq 2, J_0=0, J_1=1 \)
  
  
• Jacobsthal-Lucas Numbers: \( j_n=j_{n-1}+2j_{n-2} \\ \text{ where } n \geq 2, j_0=j_1=2\)

Today I turned 28160 days old and this number is a member of OEIS A085449:

 A085449: Horadam sequence (0,1,4,2)

The numbers indicate that the sequence is generated as follows for \(n>1\):$$H_n=2H_{n-1}+4H_{n-2}$$with \(H_0=0\), \(H_1=1\), \(r=4\) and \(s=2\)

The sequence begins: 0, 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160

The generating function is:$$ \frac{x}{1-2x-4x^2}$$The progressive ratios between successive terms approach the following number:$$ \frac{H_n}{H_{n-1}} \rightarrow 2\phi = \sqrt{5}+ 1 \text{ as }n \text{ gets larger}$$In the general case we have:$$ \frac{H_n}{H_{n-1}} \rightarrow \frac{r}{s} \phi = \frac{r}{s} (\sqrt{5}+ 1) \text{ as }n \text{ gets larger}$$Oddly, the name Horadam does not appear in the MacTutor biographies of mathematicians but Gemini provided the following summary of his life and work:

Alwyn Francis ("Horrie") Horadam (1923–2016)

Alwyn Francis ("Horrie") Horadam was a prominent Australian mathematician best known for his extensive work in number theory and for generalizing second-order linear recurrences.

Here is a comprehensive overview of his life, career, and the mathematical sequences that bear his name.

---

Early Life and Education

Horadam was born on March 22, 1923, to a family of dairy farmers in the rural settlement of Scotts Flat in the Hunter Valley of New South Wales, Australia. His dedication to education was evident early on; during the Great Depression, he traveled 110 kilometers round-trip by train every day just to attend high school in Maitland, all while managing farm duties before and after his commute.

He went on to graduate with First Class Honours in Mathematics from the University of Sydney in 1944. He later earned a BEd from the University of Melbourne and a PhD from the University of Sydney, focusing his early doctoral research on Clifford geometry in complex projective spaces.

Academic Career at UNE

Horadam spent nearly his entire 40-year academic career at the University of New England (UNE) in Armidale, New South Wales. Starting as a lecturer in 1947, he eventually progressed to Professor of Mathematics and served as the Dean of the Faculty of Science.

Beyond his research, he was widely regarded as a deeply committed and generous educator. He served on the New South Wales Junior School Mathematics Committee for 25 continuous years, co-authored several high school mathematics textbooks, and maintained an open-door policy for his university students throughout his tenure.

The Horadam Sequence

While Horadam's early interests included geometry and algebra, his most lasting mathematical legacy lies in number theory—specifically, his generalization of integer sequences. Introduced in a series of papers in the 1960s, the Horadam sequence, commonly denoted as W_n(a, b; p, q), provides a unified framework for second-order linear recurrences.

The sequence is defined by two initial conditions:

W₀ = a
W₁ = b

And the recurrence relation for n ≥ 2:

W_n = pW_n-1 - qW_n-2

(Note: Depending on the specific convention used in various literature, the recurrence is sometimes written as W_n = pW_n-1 + qW_n-2.)

By altering the four parameters (a, b, p, q), the Horadam sequence acts as a parent to many famous recursive sequences:

• Fibonacci Numbers: Generated by setting a=0, b=1, p=1, and q=-1.
• Lucas Numbers: Generated by setting a=2, b=1, p=1, and q=-1.
• Pell Numbers: Generated by setting a=0, b=1, p=2, and q=-1.

This generalization allowed mathematicians to prove broad identities and Binet-style formulas that apply universally to all these sequences at once, rather than analyzing them in isolation.

Legacy in Sequence Analysis

Horadam's generalizations paved the way for modern, systematic analysis of recurrence relations. He was a foundation and sustaining member of the Fibonacci Association and served extensively on the editorial board of the Fibonacci Quarterly. Over his lifetime, he published widely on combinatorial structures, generating functions, and sequence periodicities.

He continued his research well into his retirement, publishing his final papers in 2006. Horadam passed away in Armidale in 2016 at the age of 93, leaving behind a profound legacy in both Australian mathematical education and global number theory.

Interestingly his wife was a fellow mathematician and clearly an impressive woman:

Eleanor Mollie Horadam (née Spedding) (1921–2002)

Eleanor Mollie Horadam was a highly accomplished mathematician in her own right, with a fascinating career and a lasting legacy in Australian academia. She was the wife of Alwyn Francis Horadam.

---

Early Life and World War II

Born on June 29, 1921, in Yorkshire, England, to a suffragette mother and a pharmacist father, Mollie demonstrated early academic brilliance. She studied the Mathematical Tripos at Girton College, Cambridge, earning her BA in 1942. However, because Cambridge did not allow women to receive degrees at public graduation ceremonies at the time, she was not publicly conferred her degree until she returned on a sabbatical in 1956.

During World War II, she worked in the Stress Group at Rolls-Royce, performing stress-strain analyses on jet engines. While working by day, she took night classes in engineering at the University of London, ultimately earning a First Class Honours degree. During this time, she even outpaced early computing; when told it would take weeks to run a stress problem on the newly constructed Mark 1 computer in Manchester, she manually proved the exact mathematical solution much faster.

Move to Australia and Academic Career

Dissatisfied with post-war England's lack of opportunities and preference for promoting less-qualified men, Mollie emigrated alone to Australia in 1949 to take up a lectureship in mathematics and physics at the New England University College (which later became the University of New England, or UNE). It was here that she met fellow mathematics lecturer Alwyn Horadam, whom she married in 1950.

Mollie became a trailblazer for women in Australian academia. She successfully lobbied UNE to update its maternity policies, which allowed her to retain her lecturing position while raising their three daughters—a highly unusual achievement for the era. One of their daughters, Kathy Horadam, also went on to become a prominent Australian mathematician and Emeritus Professor at RMIT.

Mathematical Contributions

Inspired by lectures from J.E. Littlewood during her 1956 sabbatical at Cambridge, Mollie shifted her focus to number theory at the age of 35. Over the next decade, she published more than 30 research papers, primarily focusing on the number theory of generalised integers and generalised prime numbers. This extensive body of research earned her a PhD by Prior Publication from UNE in 1965, leading to her promotion to Senior Lecturer. She also authored the textbook Principles of Mathematics for Economists.

In 1970, she broke another barrier by becoming the first female Sub-Dean of a faculty (Science) at UNE. Following her academic retirement in 1982, she established a successful commercial business dealing in antique silver. She was admitted as a Fellow of the University of New England in 1995. Mollie passed away in Armidale in 2002 at the age of 80, remembered as a pioneering intellect, a community leader, and a resilient force in a predominantly male profession.

Their eldest daughter followed in her parents' footsteps:

Alwyn and Mollie Horadam had three daughters: Kathryn (Kathy), Kerry, and Alanna. They also had a total of five granddaughters and one grandson.

The most publicly known of their children is their daughter Kathryn Jennifer Horadam (born in 1951 in Armidale), who followed in her parents' footsteps to become a highly accomplished and internationally recognized mathematician.

---

Kathryn Horadam's Career and Contributions

• Education: She studied mathematics at the Australian National University, earning her bachelor's degree in 1972 and completing her PhD in 1977 with a dissertation titled The Homology of Groupnets.
• Academic Career: She built a long and distinguished career at the Royal Melbourne Institute of Technology (RMIT), where she worked for over 30 years. She became a full professor of mathematics there in 1995 and currently holds the title of Emeritus Professor.
• Research Focus: She is best known for her specialized research on Hadamard matrices and their applications in information security (keeping digital data safe). Outside of academia, she applied this expertise by working for three years with Australia's Defence Science and Technology Group.
• Publications: She authored the comprehensive text Hadamard Matrices and Their Applications, published by Princeton University Press in 2007.
• Recognition: Her impact on mathematics has been widely celebrated. She became a fellow of the Institute of Combinatorics and its Applications in 1991 and a fellow of the Australian Mathematical Society in 2001. In 2011, RMIT hosted a special international workshop on Hadamard matrices specifically in honor of her 60th birthday, with the resulting papers published in a special 2013 issue of the Australasian Journal of Combinatorics.

While Kathy pursued a high-profile academic career in mathematics like her parents, public details regarding the specific careers and lives of Kerry and Alanna are not prominently featured in public or academic records.

Zeroless Tetranacci Numbers

In a post titled, Sequences Formed By Removing Zeros, from January 2023,  I wrote that "It's interesting to consider what happens to a sequence if a certain rule is applied but with the stipulation that any zeros arising must be removed". In that post I looked at the zeroless Fibonacci sequence that falls into a repeating loop with a confirmed period of 912. 

The zeroless Tribonacci sequence falls into a much larger repeating loop with a confirmed period of 300,056,874. It reaches this cycle at index 208,666,297. However, it is not known whether the zeroless Tetranacci sequences cycles or not but, if it does, then \(s+p > 10^{10}\) where \(s\) and \(p\) are the starting index and period of the cycle, respectively.

 

 A371916: zeroless analog of tetranacci numbers.

The initial members are:

1, 1, 1, 1, 4, 7, 13, 25, 49, 94, 181, 349, 673, 1297, 25, 2344, 4339, 85, 6793, 13561, 24778, 45217, 9349, 9295, 88639, 1525, 1888, 11347, 13399, 28159, 54793, 17698, 11449, 11299, 95239, 135685, 253672, 495895, 98491, 983743, 183181, 176131, 1441546, 278461, 279319, 2175457

Figure 1 shows a plot of the first 100 terms:

Figure 1: permalink

Like the zeroless Fibonacci and Tribonacci sequences the ratio between successive terms of the zeroless Tetranacci sequence never approaches a limit. With no suppression of zeros, the following are the convergences:

• Fibonacci: $\phi = \frac{1+\sqrt{5}}{2} \text{ which is }\approx 1.61803$
• Tribonacci: the real root of $x^3 - x^2 - x - 1 = 0 \text{ which is } \approx 1.83929$
• Tetranacci: the real root of $x^4 - x^3 - x^2 - x - 1 = 0 \text{ which is }\approx 1.92756$

Primes Within Transcendental Numbers

One of the properties of the number associated with my diurnal age today, 27943, is that it is a prime formed by merging five successive digits in the decimal expansion of \(e \). Primes with this property form OEIS A104846. This got me thinking about whether all possible primes could be expressed by merging successive digits in the decimal expansions of all transcendental numbers like \(e\) and \( \pi \).  

However, as Grok explained:

No, it cannot be proven that the decimal expansions of transcendental numbers like \( \pi \) or \(e\) contain representations of all prime numbers through progressive concatenations of consecutive digits. This interpretation hinges on whether every possible finite sequence of digits—including the decimal representations of all primes—appears as a substring somewhere in their infinite, non-repeating expansions. While this would be true if \( \pi \) and \(e\) are normal numbers (where every digit sequence of length \(k\) appears with equal frequency \(1/10^k\) in the limit), the normality of \( \pi \) and \(e\) remains an open problem in mathematics. 

If numbers like \( \pi \) and \(e\) could be proven to be normal then not only would every prime occur but it would occur an infinite number of times. It's interesting to look at 27943 and see how many times it appears in the first million digits of \(e\). Here are the initial positions for the "2" (permalink):

166, 46606, 217029, 234391, 248347, 258504, 483659, 832027, 941722

For \( \pi \), it takes a while before the first occurrence but afterwards the occurrences of 27943 are more frequent (permalink):

94943, 146738, 216793, 253032, 283291, 290571, 302660, 350882, 358637, 520562, 632389, 661691, 715324, 805161, 843112, 890626

I wondered if there was any difference between algebraic numbers like \( \phi \), the golden ratio, and transcendental numbers in regards to being able to represent all primes as concatenations of successive digits. Here's was Grok's response:

No, it cannot be proven that the decimal expansions of algebraic irrational numbers like the golden ratio (\( \phi  \approx 1.6180339887 \dots \)) contain representations of all prime numbers through progressive concatenations of consecutive digits, for essentially the same reasons as with transcendentals like \( \pi \) and \(e\). Their normality in base 10 remains unproven. 

Just as with π and e, the golden ratio's decimal expansion is infinite and non-repeating (since it's irrational), but this doesn't guarantee it contains all sequences. Empirical checks of millions of digits suggest randomness consistent with normality, but no proof exists—not even that all digits 0-9 appear infinitely often. It's widely conjectured that φ (a quadratic irrational algebraic number) is normal in base 10, mirroring beliefs about π and e. Proving this would be a major result in number theory.

There's a specific conjecture (dating back to Borel in 1909, refined later) that every irrational algebraic number is absolutely normal (normal in every integer base \(\geq 2)\). This includes \( \phi, \sqrt{2}, \sqrt{3} \), and others. No counterexamples exist, but crucially, no irrational algebraic has ever been proven normal in any base. In contrast, for transcendentals, normality isn't conjectured for all—only for certain "natural" ones like \( \phi \) and \(e\)—while many non-normal transcendentals are known (e.g., Liouville numbers designed to lack certain digits). 

Proven examples: We have constructed transcendentals proven to be normal, such as Champernowne's constant ( 0.123456789101112 ...)  or the Copeland–Erdős constant (formed by concatenating primes: 0.235711131719... ). These do contain every finite digit sequence infinitely often, including all primes. No such proven-normal algebraic irrationals exist, partly because algebraics satisfy polynomial equations, which might impose subtle constraints on their expansions (though the conjecture suggests not). 

In summary, algebraic irrationals like \( \phi \) aren't fundamentally "different" in the sense that the core question—whether all primes appear as substrings—remains unproven and conjectured true, just like for \( \phi\) and \(e\). The main distinction lies in the broader landscape: Algebraics have a stronger universal conjecture without exceptions or proven cases, while transcendentals show a mix of proven normal and non-normal examples. If the conjectures hold, then yes, \( \phi \)'s expansion would contain all primes, but we can't prove it yet.

So at what positions does the prime 27943 appear in the decimal expansion of \( \phi \). Here are the locations (permalink):

33280, 127164, 129935, 529433, 597796, 795399, 971620

Which is Larger?

The question was posed in a YouTube video as to which was larger: \(3.14^{\pi}\) or \( \pi^{3.14}\) ? The use of a calculator was not allowed. The solution provided in the video is interesting.$$ \begin{align} &3.14^{\pi} \text{ versus } \pi^{3.14} \\ &\text{raise each to the } 1/3.14 \text{ power} \\ &3.14^{\pi /3.14} \text{ versus } \pi^{3.14/3.14} \\ &3.14^{\pi/3.14} \text{ versus } \pi  \\ & \text{raise each to the } 1/ \pi \text{ power}  \\ &3.14^{\pi/(3.14 \pi)} \text{ versus } \pi ^{1/\pi} \\   &3.14^{1/3.14} \text{ versus } \pi ^{1/\pi} \end{align} $$Now consider the function \(y=x^{1/x}\) which we will modify as follows:$$ \begin{align} y &=x^{1/x} \\ &= e^{\ln{x^{1/x}}} \\ &=e^{1/x \ln{x}} \end{align}$$Let's find its first derivative in order to determine stationary values:$$ \begin{align} \frac{dy}{dx} &=e^{1/x \ln{x}} \frac{d}{dx} (1/x \ln{x})\\ &= \frac{e^{1/x \ln{x}}}{x^2}(1-\ln{x})  \end{align} $$the numerator and denominator on the left are always positive so it comes down to:$$ \begin{align} 1-\ln{x} = 0  \text{ when } x=e \\  \\ \text{if } x<e, dy/dx > 0 \text{ and if } x>e, dy/dx < 0 \end{align}$$Thus there is a maximum turning point at \(x=e\) and since, after this point, \(y\) decreases as \(x\) increases, we must have:$$3.14^{1/3.14} > \pi^{1/\pi} \text{ since } 3.14 < \pi$$This in turn means that by reversing our original transformation, we have:$$3.14^{\pi} > \pi^{3.14}$$Figure 1 shows what the graph of \(y=x^{1/x}\) look like.

Figure 1: note the gradual descent as \(x\) gets larger

The approximate values of our two expressions are as follows:$$ \begin{align} 3.14^{\pi}&\approx 36.4041195357888 \\ \pi^{3.14} &\approx 36.3957438848941 \end{align}$$The solution to this problem also settles the more general issues surrounding when \(a^b\) is larger or smaller than \(b^a\) with \(a,b>1\). Let's consider the problem of:$$a^b \text{ versus } b^a \text{ for the particular case of }2024^{2025} \text{ versus } 2025^{2004}$$We know from earlier that this comparison reduces to:$$ \begin{align} &a^{1/a} \text{ versus } b^{1/b} \text{ or } 2024^{1/2024} \text{ versus } 2025^{1/2005} \\  &\text{and if }a>e, b>e \text{ and } a<b \text{ then:} \\ &\text{because } 2024 < 2025 \text{ we have:}\\ &2024^{2025} > 2025^{2004} \end{align} $$ However, what about:$$ \phi^{1.618} \text{ versus } 1.618^{\phi}$$Here the problem reduces to:$$\phi^{1/\phi} \text{ versus } 1.618^{1/1.618}$$Here \(\phi<e\) and \(1.618<e\) and \(1.618 < \phi\), so:$$ 1.618^{1/1.618} < \phi ^{1/\phi} \text{ and } 1.618^{\phi}<\phi^{1.618}$$Double checking we find that:$$ \begin{align} 1.618^{\phi} \approx 2.17838352536880\\ \phi^{1.618} \approx 2.17842193783803 \end{align}$$If the values of \(a\) and \(b\) fall on either side of \(e\) then the previous methods don't work. I'll leave this problem for another post possibly as I've spent enough time on this.

An Interesting Intersection

I've never encountered an intersection of the arithmetic derivative and the totient function before until looking at one of the properties associated with the number 27620, my diurnal age yesterday. This number has an interesting property that qualifies it for membership in OEIS A352332:

A352332: numbers \(k\) for which \(k = \phi(k') + \phi(k'') \), where \( \phi \) is the Euler totient function (A000010), \(k'\) is the arithmetic derivative of \(k\) (A003415) and \(k''\) is the second arithmetic derivative of \(k\) (A068346).

Let's look at 27620 and its arithmetic derivatives (denoted by ' and '') and their totients (denoted by \( \phi \) ). Firstly:$$ \begin{align} 27620 &= 2^2 \times 5 \times 1381\\  (27620)' &=33164 \\(27620)'' &= (33164)'\\ &=33168 \\ \phi(33164) &= 16580 \\ \phi(33168) &= 11040 \\ 27620 &= 16580 + 11040 \end{align} $$Up to 40000, the members of this sequence are 4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620 and 37460 (permalink). Clearly, numbers satisfying these criteria are few and far between.

This got me thinking about whether there were any numbers that satisfied the criterion that they were simply the sum of their first and second arithmetic derivatives, ignoring the totient function. There are only three numbers in the range up to 100,000 that satisfy. They are 6, 42, 15590 and 47058. Here is the breakdown:$$ \begin{align} 6 &= 5 + 1\\42 &= 41 + 1\\15590 &= 10923 + 466\\47058 &= 47057 + 1 \end{align}$$It can be noted that in all but 15590, the first derivative yields a prime and thus the second derivative is 1.

What if, instead of the totient, we use the sum of the divisors (\(\sigma\)) of the number? In the range up to 100,000, no numbers satisfy but if we use the sum of the PROPER divisors, then a single number qualifies and that is 50 (permalink). Let's look more closely:$$ \begin{align} 50 &= 2 \times 5^2 \\ (50)' &= 45 \\ \sigma(45) -45&= 78 - 45 \\ &=33\\(50)'' &=(45)'\\ &=39\\ \sigma(39)-39 &= 56-39\\ &= 17 \\50 &= 33 +17 \end{align} $$It should be noted that numbers like 50, 15590 and 27620 with the specific properties that have been mentioned in this post retain these properties in bases other than 10. In other words, the properties are base independent.

An Interesting Prime

Being born on the 3rd April 1949, my date of birth is often represented as 3 - 4 - 49. These numbers when concatenated form the prime number 3449. I was reminded of this number because of the factorisation of the number associated with my diurnal age today, 27592.$$27592 = 2^3 \times 3449 = 8 \times 3449$$So today my life can be divided into exactly eight equal parts, each of them 3449 days long which is about 9.44 years. The previous multiple$$7 \times 3449 = 24143$$occurred on May 10th 2015 when I was still working at the Shanghai Singapore International School. The next multiple$$9 \times 3449 = 31041$$ will fall on March 29th 2034, shortly before my 84th birthday (if I make it that far).

3449 forms the initial prime of a Cunningham chain of the first type with length exactly 3 and so: $$ 3449 \rightarrow 2 \times 3449 + 1 = 6889 \text{ (prime)} \\ 6889 \rightarrow 2 \times 6889 +1 = 13799 \text{ (prime)}$$Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes \( \textit{p} \) such that \(p^2+p-1\) and \(p^2+p+1\) are twin primes.

For 3449, the twin primes are \(11899049\) and \(11899051\). 

3449 is also a home prime with a homeliness of 3 because:$$ \begin{align} 611 &= 13 \times 47 \rightarrow 1347\\ 1347 &= 3 \times 449 \rightarrow 3449 \end{align}$$Not all primes are home primes of course. Take 613 as an example of a prime that is not a home prime because it cannot be formed by the concatenation of the prime factors of any number (the prime factors need to be concatenated in ascending order).

3449 is also a member of OEIS A153116:

A153116: primes \(p\) such that \(p^2 +12\) and \(p^2-12\) are also primes.

Here the two primes are \(11895589\) and \(11895613\). Additionally:$$ \text{period of}\frac{1}{3449}=\frac{3449-1}{8} = 431$$This property qualifies 3449 for membership of OEIS A056213:

A056213: primes \(p\) for which the period of reciprocal = \(\dfrac{p-1}{8}\).

3449 is a Sophie Germain prime because:$$2 \times 3449+1=6899 \text{ is prime}$$3449 also features in so-called "Golden Semiprimes" and this qualifies it for membership in OEIS A108544:

A108544: primes that are factors of distinct golden semiprimes (A108540).

I posted about these types of semiprimes in Semiprime Factor Ratios way back on the 26th August 2016. In that post I said that:

A golden semiprime is a number that factors to:

• \(p \times q\) where \(p\) and \(q\) are prime
• \( | \,p \, \phi - q \,| <1\) where \(\phi=\dfrac{\sqrt{5}+1}{2} \)

In the case of 3449, it is the \(q\) and \(p=2131\) and the golden semiprime is:$$7349819=2131 \times 3449$$The OEIS mentions 314 sequences in which 3449 makes an appearance and I've only dealt with a few of them here. However, I see 3449 as an important number in my life and didn't want its current occurrence to pass unnoticed.

Properties of Concatenated Numbers

Let's split a number in two parts and then consider the relationship that these two parts bear to the number of which both are a part. Now I've done this before as described in a post titled Energetic Numbers. In THAT post, I looked at numbers that belong to OEIS  A055480:

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

In THIS post I'll be looking at two different types of relationships. Let's consider 27573 and break it into two parts: 275 and 73. It turns out that my diurnal age today (27573) has an interesting property and it involves the totient function, commonly referred to as Euler's phi function or simply the phi function. We find that:$$ \begin{align}  \phi(27573) &= 1440 \\ \phi(275) \times \phi(73) &= 200 \times 72\\ &=1440 \end{align} $$Numbers like this belong to OEIS A147619 :

 A147619             Numbers \(n\)  = \(a \,| \, b\) such that \( \phi(n) = \phi(a) \times \phi(b) \)
                                where the symbol | represents concatenation

The initial members of this sequence are:

78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598, 2967, 3168, 3762, 4758, 5775, 5796, 7800, 7875, 7917, 8217, 8970, 9180, 9576, 11790, 13650, 13662, 13875, 13896, 14391, 17760, 18564, 18858, 19812, 20097, 25740, 25935, 25974, 25980, 27573, 28776, 28779, 29670, 31680, 33165, 35919, 37620

Let's look at the breakdown for the numbers between 27573 and 37620:

• 275 | 73  -->  27573
  
  
•  287 | 76  -->  28776
  
  
•  287 |  79  -->  28779
  
  
•  29 | 670  -->   29670
  
  
•  31 | 680  -->   31680
  
  
•  331 | 65  -->   33165
  
  
•  35 | 919  -->   35919
  
  
•  37 | 620  -->   37620

A similar thing can be done with the sigma function that tallies the sum of the divisors of a number. Once again we break the number into two parts and then use the sigma function to connect those two parts to the whole. There is no OEIS function this time around but the property of the numbers being considered is as follows:

 Numbers \(n\)  = \(a \,| \, b\) such that \( \sigma(n) = \sigma(a) \times \sigma(b) \)
where the symbol | represent concatenation

The 78 numbers satisfying this condition between 27000 and 40000 are (permalink):

27175, 27298, 27418, 27445, 27500, 28018, 28195, 28750, 28798, 28978, 29038, 29058, 29098, 29278, 29395, 29398, 29500, 29875, 29875, 29950, 29950, 29980, 30498, 30775, 30788, 30989, 31750, 31795, 31918, 32595, 33175, 33238, 33298, 33725, 34189, 34555, 34557, 34795, 34975, 35338, 35395, 35578, 35585, 35670, 35818, 35975, 36178, 36238, 36350, 36398, 36426, 36775, 37138, 37241, 37678, 37750, 37798, 38247, 38370, 38518, 38570, 38638, 38750, 38826, 38856, 38877, 38940, 38975, 39118, 39425, 39478, 39500, 39754, 39805, 39898, 39976, 39980, 39992

Table 1 shows the breakdown for numbers between 27000 and 30000:

Table 1

Let's look at the first number 27175 where we break the number into 271 and 75 and we have:$$ \begin{align} \sigma(27175) &= 33728\\ \sigma(271) \times \sigma(75) &= 272 \times 124\\ &= 33728 \end{align}$$There are other possible concatenations involving more than two parts or other functions and properties of numbers so this post is merely a sample.

A Special Number Plate

Last night I noticed an unusual car number plate. It was 432 432.  This is a customised number plates as standard number plates follow an AAA 000 pattern, that is three uppercase letters followed by three digits. Presumably the number 432 was of some significance to the person who purchased the plates. This got me thinking about what is special about the number 432.

432 432

PROPERTY 1

The first property of interest is that it's wedged between two prime numbers: 431 and 433. Thus we have:$$432 = \frac{431+433}{2}$$This qualifies it for membership in OEIS A014574:

A014574

Average of twin prime pairs.

The initial members of this sequence are:

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

PROPERTY 2

The next property of interest is that it's the sum of two cubes. Specifically:$$432=6^3+6^3$$This property qualifies it for membership in OEIS A003325:

A003325

Numbers that are the sum of 2 positive cubes.

The initial members of this sequence are:

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

PROPERTY 3

The next property is that it's the sum of the totients of the first 37 numbers: $$432=\sum_{n=1}^{37} \phi(n)$$This qualifies it for membership in OEIS A002088:

A002088

Sum of totient function: \( \displaystyle{\text{a}(n) = \sum_{k=1}^n \phi(k) } \)

The initial members of the sequence are:

0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

PROPERTY 4

  

Another property makes it a member of OEIS A033833:

A033833

Highly factorizable numbers: numbers with a record number of proper factorizations.

432 turns out to have 56 possible factors which are:

[2, 2, 2, 2, 3, 3, 3], [2, 2, 2, 2, 3, 9], [2, 2, 2, 2, 27], [2, 2, 2, 3, 3, 6], [2, 2, 2, 3, 18], [2, 2, 2, 6, 9], [2, 2 , 2, 54], [2, 2, 3, 3, 3, 4], [2, 2, 3, 3, 12], [2, 2, 3, 4, 9], [2, 2, 3, 6, 6], [2, 2, 3, 36], [2, 2, 4, 27], [2, 2, 6, 18], [2, 2, 9, 12], [2, 2, 108], [2, 3, 3, 3, 8], [2, 3, 3, 4, 6], [2, 3, 3, 24], [2, 3, 4, 18], [2, 3, 6, 12], [2, 3, 8, 9], [2, 3, 72], [ 2, 4, 6, 9], [2, 4, 54], [2, 6, 6, 6], [2, 6, 36], [2, 8, 27], [2, 9, 4], [2, 12, 18], [2, 216], [3, 3, 3, 4, 4], [3, 3, 3, 16], [3, 3, 4, 12], [3, 3, 6, 8], [3, 3, 48], [3, 4, 4, 9], [3, 4, 6, 6], [3, 4, 36], [3, 6, 24], [3, 8, 18], [3, 9, 16], [3, 12, 12], [3, 144], [4, 4, 27], [4, 6, 18], [4, 9, 12], [4, 108], [6, 6, 12], [6, 8, 9], [6, 72], [8, 54], [9, 48], [12, 36], [16, 27], [18, 24]

It can be noted that 666 makes its appearance since 432 = 2 x 6 x 6 x 6.

The initial members of the sequence are:

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

PROPERTY 5

Another property of 432 is that it's the difference between the squares of two successive primes. Specifically$$ \begin{align} 432&=109^2-107^2\\&=(109+107) \times (109-107)\\&=216 \times 2\\ &=2^4 \times 3^3 \end{align} $$This qualifies 432 for inclusion in OEIS A069482:

A069482

a(\(n\)) = (prime(\(n\)+1))\(^2\) - (prime(\(n\)))\(^2\)

The initial members of the sequence are:

5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848

I'll stop there as I think I've shown that 432 has at least five interesting properties but of course there are many more. There are actually 4018 entries for this number in the OEIS.