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
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:
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| 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:
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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.
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