When Sequences Overlap

I've written in previous posts about what I've called the Factors Sequence (with and without multiplicity) and the Divisors Sequence:

• Number's Divisors to Sequence Algorithm
• Number's Factors to Sequence Algorithm 2
• Number's Factors to Sequence Algorithm 1

What I noticed about the number (28268) associated with my diurnal age today is that for the Factors Sequence with multiplicity and the Divisors Sequence both terminate where they began. Here is the Factors Sequence: 28268, 7067, 14134, 42402, 169608, 28268. Figure 1 shows a plot of the points with a logarithmic vertical axis.

Figure 1

Here is Divisors Sequence: 28268, 339216, 13568640, 106005, 1696080, 21201, 169608, 5427456, 390776832, 2035296, 28268. Figure 2 shows a plot of the points with a logarithmic vertical axis:

Figure 2

Closer inspection of both sequences shows that they share a common number apart from 28268. The number is 169608 and Figure 3 shows both plots on the same graph.

Figure 3: permalink

In the range up 40000, there are 1779 numbers with these two properties:

• the factors sequence with multiplicity and the divisors sequence terminate where they started
• the two sequences share a common member different from the start and end values

Here are the numbers (permalink) from 28000 to 40000:

28001, 28012, 28019, 28036, 28056, 28124, 28132, 28148, 28163, 28204, 28244, 28248, 28252, 28268, 28292, 28307, 28324, 28348, 28364, 28372, 28388, 28396, 28444, 28468, 28492, 28499, 28532, 28536, 28564, 28588, 28612, 28628, 28643, 28676, 28684, 28724, 28732, 28776, 28793, 28796, 28804, 28868, 28892, 28901, 28924, 28949, 28961, 28964, 28968, 28979, 28996, 29021, 29033, 29044, 29064, 29084, 29092, 29108, 29116, 29123, 29156, 29164, 29201, 29212, 29243, 29252, 29256, 29276, 29308, 29339, 29356, 29372, 29428, 29444, 29452, 29453, 29468, 29483, 29492, 29516, 29548, 29564, 29588, 29612, 29636, 29684, 29692, 29708, 29756, 29764, 29784, 29788, 29812, 29832, 29852, 29873, 29884, 29928, 29932, 29972, 30004, 30052, 30072, 30076, 30124, 30172, 30203, 30264, 30269, 30284, 30323, 30347, 30388, 30389, 30404, 30408, 30436, 30449, 30452, 30467, 30476, 30504, 30508, 30524, 30532, 30539, 30548, 30552, 30604, 30644, 30652, 30689, 30716, 30773, 30788, 30803, 30836, 30844, 30884, 30916, 30956, 30988, 31004, 31052, 31076, 31084, 31124, 31132, 31139, 31148, 31196, 31204, 31228, 31244, 31252, 31253, 31259, 31276, 31324, 31348, 31396, 31436, 31444, 31469, 31484, 31512, 31547, 31564, 31588, 31612, 31636, 31649, 31652, 31721, 31756, 31793, 31828, 31844, 31868, 31876, 31916, 31924, 31964, 31988, 31992, 31996, 32003, 32009, 32012, 32084, 32088, 32092, 32108, 32132, 32136, 32141, 32188, 32204, 32228, 32232, 32252, 32284, 32308, 32332, 32376, 32381, 32424, 32476, 32507, 32516, 32524, 32548, 32561, 32564, 32568, 32572, 32573, 32596, 32603, 32612, 32633, 32636, 32692, 32712, 32756, 32788, 32789, 32804, 32812, 32828, 32843, 32852, 32908, 32933, 32955, 32987, 32996, 33004, 33028, 33044, 33053, 33068, 33096, 33107, 33116, 33196, 33236, 33284, 33288, 33308, 33332, 33347, 33356, 33364, 33384, 33388, 33404, 33428, 33432, 33436, 33461, 33484, 33521, 33528, 33532, 33569, 33596, 33604, 33628, 33644, 33652, 33668, 33672, 33713, 33749, 33764, 33773, 33809, 33812, 33836, 33864, 33884, 33892, 33916, 33932, 33941, 33956, 33964, 33988, 34008, 34012, 34028, 34036, 34076, 34124, 34196, 34204, 34228, 34244, 34253, 34268, 34316, 34348, 34372, 34412, 34444, 34468, 34484, 34532, 34556, 34584, 34604, 34612, 34628, 34636, 34667, 34732, 34844, 34868, 34913, 34949, 34968, 34972, 34996, 35036, 35068, 35069, 35081, 35092, 35108, 35164, 35188, 35204, 35236, 35252, 35256, 35339, 35363, 35372, 35404, 35428, 35448, 35492, 35516, 35524, 35564, 35573, 35596, 35612, 35636, 35668, 35684, 35708, 35756, 35788, 35812, 35908, 35924, 35932, 35933, 35956, 35963, 35993, 36024, 36068, 36076, 36083, 36092, 36124, 36148, 36168, 36188, 36212, 36284, 36292, 36299, 36308, 36312, 36316, 36332, 36353, 36356, 36388, 36404, 36408, 36428, 36452, 36467, 36476, 36484, 36501, 36524, 36572, 36596, 36629, 36668, 36676, 36683, 36696, 36716, 36761, 36772, 36788, 36821, 36844, 36868, 36888, 36892, 36916, 36923, 36929, 36932, 36984, 36988, 37012, 37013, 37036, 37049, 37052, 37084, 37148, 37156, 37181, 37196, 37204, 37228, 37252, 37253, 37316, 37388, 37412, 37464, 37516, 37532, 37547, 37556, 37628, 37708, 37796, 37804, 37848, 37853, 37876, 37924, 37948, 37972, 38036, 38068, 38092, 38108, 38116, 38136, 38184, 38189, 38201, 38212, 38228, 38252, 38276, 38284, 38308, 38333, 38356, 38372, 38396, 38428, 38444, 38453, 38468, 38472, 38501, 38548, 38564, 38588, 38603, 38636, 38668, 38669, 38684, 38692, 38732, 38747, 38764, 38804, 38812, 38828, 38852, 38861, 38867, 38873, 38908, 38924, 38933, 39028, 39044, 39052, 39089, 39092, 39107, 39144, 39172, 39188, 39192, 39196, 39233, 39236, 39308, 39323, 39336, 39364, 39388, 39412, 39419, 39432, 39452, 39476, 39521, 39524, 39569, 39572, 39576, 39596, 39624, 39652, 39668, 39748, 39772, 39779, 39812, 39827, 39836, 39844, 39864, 39908, 39916, 39932, 39953, 39956, 39964, 39988, 39989

Proth Numbers

My first dedicated post about Proth numbers dates back to January of 2020. I also mention them in Sierpinski Numbers (June 2026) and Riesel and Sierpinski Numbers (April 2021). Let's review the definition of a Proth number:

A Proth number is any whole number written in the form \(k \cdot 2^n + 1\), where \(k\) is an odd number, \(n\) is a positive whole number, and \(2^{n}\) is greater than \(k\).

I've never listed all the Proth numbers in the range up to 40000 so I got Gemini to write a program to do this and here is the result (permalink):

Proth Numbers:

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409, 1473, 1537, 1601, 1665, 1729, 1793, 1857, 1921, 1985, 2049, 2113, 2177, 2241, 2305, 2369, 2433, 2497, 2561, 2625, 2689, 2753, 2817, 2881, 2945, 3009, 3073, 3137, 3201, 3265, 3329, 3393, 3457, 3521, 3585, 3649, 3713, 3777, 3841, 3905, 3969, 4033, 4097, 4225, 4353, 4481, 4609, 4737, 4865, 4993, 5121, 5249, 5377, 5505, 5633, 5761, 5889, 6017, 6145, 6273, 6401, 6529, 6657, 6785, 6913, 7041, 7169, 7297, 7425, 7553, 7681, 7809, 7937, 8065, 8193, 8321, 8449, 8577, 8705, 8833, 8961, 9089, 9217, 9345, 9473, 9601, 9729, 9857, 9985, 10113, 10241, 10369, 10497, 10625, 10753, 10881, 11009, 11137, 11265, 11393, 11521, 11649, 11777, 11905, 12033, 12161, 12289, 12417, 12545, 12673, 12801, 12929, 13057, 13185, 13313, 13441, 13569, 13697, 13825, 13953, 14081, 14209, 14337, 14465, 14593, 14721, 14849, 14977, 15105, 15233, 15361, 15489, 15617, 15745, 15873, 16001, 16129, 16257, 16385, 16641, 16897, 17153, 17409, 17665, 17921, 18177, 18433, 18689, 18945, 19201, 19457, 19713, 19969, 20225, 20481, 20737, 20993, 21249, 21505, 21761, 22017, 22273, 22529, 22785, 23041, 23297, 23553, 23809, 24065, 24321, 24577, 24833, 25089, 25345, 25601, 25857, 26113, 26369, 26625, 26881, 27137, 27393, 27649, 27905, 28161, 28417, 28673, 28929, 29185, 29441, 29697, 29953, 30209, 30465, 30721, 30977, 31233, 31489, 31745, 32001, 32257, 32513, 32769, 33025, 33281, 33537, 33793, 34049, 34305, 34561, 34817, 35073, 35329, 35585, 35841, 36097, 36353, 36609, 36865, 37121, 37377, 37633, 37889, 38145, 38401, 38657, 38913, 39169, 39425, 39681, 39937

Total Proth numbers between 1 and 40000: 283

Percentage of total range: 0.7075%

I asked Gemini whether there was any special name given to Proth-like numbers of the form \(k \cdot 2^n - 1\) and here was its response:

Yes, numbers of the form $k \cdot 2^n - 1$ (where $k$ is an odd positive integer and $2^n > k$) are formally known as Proth numbers of the second kind.

If a number in this sequence is prime, it is appropriately called a Proth prime of the second kind.

Depending on the specific value assigned to $k$, these numbers also branch into several famously named subsets in number theory: 

• Mersenne numbers: when $k = 1$, the formula simplifies to the widely studied $2^n - 1$. 

• Thâbit (or Thâbit ibn Kurrah) numbers: when $k = 3$, yielding $3 \cdot 2^n - 1$. 

• Woodall numbers: When $k$ is equal to the exponent $n$, generating numbers of the form $n \cdot 2^n - 1$. Note that this category drops the strict $2^n > k$ inequality rule.

This $k \cdot 2^n - 1$ formula is also the foundational equation for Riesel numbers. A Riesel number is not the output of the formula, but rather a specific odd integer $k$ that guarantees $k \cdot 2^n - 1$ will be composite for every possible integer $n$. This makes Riesel numbers the direct mathematical counterpart to Sierpiński numbers, which apply the exact same always-composite rule to the standard Proth sequence ($k \cdot 2^n + 1$).

Of course the Proth numbers of the second kind will be 2 less the Proth numbers listed earlier but I'll list them here for completeness (permalink):

Proth Numbers of the Second Kind:

1, 3, 7, 11, 15, 23, 31, 39, 47, 55, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 287, 319, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, 831, 863, 895, 927, 959, 991, 1023, 1087, 1151, 1215, 1279, 1343, 1407, 1471, 1535, 1599, 1663, 1727, 1791, 1855, 1919, 1983, 2047, 2111, 2175, 2239, 2303, 2367, 2431, 2495, 2559, 2623, 2687, 2751, 2815, 2879, 2943, 3007, 3071, 3135, 3199, 3263, 3327, 3391, 3455, 3519, 3583, 3647, 3711, 3775, 3839, 3903, 3967, 4031, 4095, 4223, 4351, 4479, 4607, 4735, 4863, 4991, 5119, 5247, 5375, 5503, 5631, 5759, 5887, 6015, 6143, 6271, 6399, 6527, 6655, 6783, 6911, 7039, 7167, 7295, 7423, 7551, 7679, 7807, 7935, 8063, 8191, 8319, 8447, 8575, 8703, 8831, 8959, 9087, 9215, 9343, 9471, 9599, 9727, 9855, 9983, 10111, 10239, 10367, 10495, 10623, 10751, 10879, 11007, 11135, 11263, 11391, 11519, 11647, 11775, 11903, 12031, 12159, 12287, 12415, 12543, 12671, 12799, 12927, 13055, 13183, 13311, 13439, 13567, 13695, 13823, 13951, 14079, 14207, 14335, 14463, 14591, 14719, 14847, 14975, 15103, 15231, 15359, 15487, 15615, 15743, 15871, 15999, 16127, 16255, 16383, 16639, 16895, 17151, 17407, 17663, 17919, 18175, 18431, 18687, 18943, 19199, 19455, 19711, 19967, 20223, 20479, 20735, 20991, 21247, 21503, 21759, 22015, 22271, 22527, 22783, 23039, 23295, 23551, 23807, 24063, 24319, 24575, 24831, 25087, 25343, 25599, 25855, 26111, 26367, 26623, 26879, 27135, 27391, 27647, 27903, 28159, 28415, 28671, 28927, 29183, 29439, 29695, 29951, 30207, 30463, 30719, 30975, 31231, 31487, 31743, 31999, 32255, 32511, 32767, 33023, 33279, 33535, 33791, 34047, 34303, 34559, 34815, 35071, 35327, 35583, 35839, 36095, 36351, 36607, 36863, 37119, 37375, 37631, 37887, 38143, 38399, 38655, 38911, 39167, 39423, 39679, 39935

Total Proth numbers of the second kind between 1 and 40000: 283

Percentage of total range: 0.7075%

Surface Area of a Regular Dodecadhedron

One of the properties of the number associated with my diurnal age today is that it represents the surface area (rounded to the nearest whole number) of a dodecahedron with an edge of 37 units. The formula and result is shown below where \(n\) represents edge length:$$ \begin{align} \text{Surface Area } &= 3 \times n^2 \times \sqrt{{25} + 10 \times \sqrt{5}} \\ &= 3 \times 37^2 \times \sqrt{25 + 10 \times \sqrt{5}} \text{ when }n=37\\ \\&= 28264.0027368755 \dots \\ \\ &\approx 28264 \end{align}$$What's special about the dodecahedron with an edge of 37 units is how close its exact surface area approaches a whole number. The table below shows the results for edges from 1 to 44:

Edge   With Decimal   Rounded

  1      20.645729      21
  2      82.582915      83
  3      185.81156      186
  4      330.33166      330
  5      516.14322      516
  6      743.24624      743
  7      1011.6407      1012
  8      1321.3266      1321
  9      1672.3040      1672
  10     2064.5729      2065
  11     2498.1332      2498
  12     2972.9849      2973
  13     3489.1282      3489
  14     4046.5628      4047
  15     4645.2890      4645
  16     5285.3066      5285
  17     5966.6156      5967
  18     6689.2161      6689
  19     7453.1081      7453
  20     8258.2915      8258
  21     9104.7664      9105
  22     9992.5327      9993
  23     10921.591      10922
  24     11891.940      11892
  25     12903.581      12904
  26     13956.513      13957
  27     15050.736      15051
  28     16186.251      16186
  29     17363.058      17363
  30     18581.156      18581
  31     19840.545      19841
  32     21141.226      21141
  33     22483.199      22483
  34     23866.462      23866
  35     25291.018      25291
  36     26756.865      26757
  37     28264.003      28264
  38     29812.432      29812
  39     31402.154      31402
  40     33033.166      33033
  41     34705.470      34705
  42     36419.066      36419
  43     38173.953      38174
  44     39970.131      39970

Clearly the dodecahedron with edge of 37 units is a clear winner as it has a surface area that differs by only about 0.003 from its surface area when rounded to the nearest whole number.

PRIME + and NON-PRIME - Improved Format

As with the EVEN + and ODD - and EVEN - and ODD + recursive algorithms, I've taken the original 393 page data document and converted it to a 83 page PDF that is more reader friendly. Here some excerpts.

Figure 1

Figure 2

Figure 3

Table 1

Table 2

Table 3

Table 4

Temperature Conversion

I came across this interesting snippet of information today:

I was interested in why this interchange should be possible and so I asked Gemini. Here was its response:

The trick works because the conversion slope between Celsius and Fahrenheit:$$\frac{9}{5} = 1.8$$closely mirrors how reversing a two-digit number changes its numerical value in base-10 arithmetic, combined with simple integer rounding.

---

The Exact Arithmetic

The standard conversion formula from Celsius ($C$) to Fahrenheit ($F$) is:

$$F = \frac{9}{5}C + 32 = 1.8C + 32$$

Every time $C$ increases by 12°C, $F$ increases by:

$$\Delta F = 1.8 \times 12 = 21.6$$

which is approximately 22°F.

Evaluating each step from the image reveals how closely the exact arithmetic matches the digit reversal:

• 04°C: $1.8(4) + 32 = 7.2 + 32 = 39.2$ → 39.2°F ≈ 40°F 
  (reversed digits: 04 → 40)
• 16°C: $1.8(16) + 32 = 28.8 + 32 = 60.8$ → 60.8°F ≈ 61°F
  (reversed digits: 16 → 61)
• 28°C: $1.8(28) + 32 = 50.4 + 32 = 82.4$ → 82.4°F ≈ 82°F
  (reversed digits: 28 → 82)
• 40°C: $1.8(40) + 32 = 72.0 + 32 = 104.0$ → 104°F
  (reversed digits: 40 → 04, plus a leading 1)
• 52°C: $1.8(52) + 32 = 93.6 + 32 = 125.6$ → 125.6°F ≈ 126°F ≈ 125°F
  (reversed digits: 52 → 25, plus a leading 1)

---

Why Digits Invert in Steps of 12

Let a two-digit Celsius value be represented by digits $a$ and $b$:

$$C = 10a + b$$

Its digit reversal $R(C)$ is:

$$R(C) = 10b + a$$

If you add $12$ to $C$, the tens digit increases by $1$ and the units digit increases by $2$ (assuming no carryover):

$$C' = 10(a+1) + (b+2)$$

The reversal of this new value is:

$$R(C') = 10(b+2) + (a+1) = (10b + a) + 21 = R(C) + 21$$

Because reversing digits after adding $12$ increases the reversed value by $21$, and the physical temperature conversion rate is $1.8 \times 12 = 21.6$, the rounded whole-number sequences track together with remarkable accuracy across the everyday range from 0°C to 50°C.

Super Sphenic Numbers

The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384I and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.

ODD - and EVEN + Improved Format

Some time ago I got Gemini to create a 197 page document that identifies attractors and vortices arising from the recursive ODD - and EVEN + algorithm. It also lists their number of captives. Today I got Gemini to summarise and reformat this information so that it is more readable using the exact same template it created for my August 15th post titled ODD + and EVEN - Improved Format. That 142 page document can be located here. Here are some excerpts:

Figure 1

Figure 2

Table 1

Table 2

Table 3

Table 4

Notice that while 8987 has the record number of captives under the ODD - and EVEN + recursive algorithm (in the range up to 40000), it has ZERO captives under the ODD + and EVEN - recursive algorithm. Conversely, while 38013 is a vortical in the mighty vortex {38013, 38012, 38006, 37995, 38028} with 564 captives under the ODD + and EVEN - recursive algorithm, it is a mere captive of the attractor 38050 under the ODD - and EVEN + recursive algorithm.