The number associated with my diurnal age today, \( \textbf{28284} \), has a property that grants it membership in OEIS A338902:
I got Gemini to write a program to find the number of suitable partitions for integers from 1 to 50 displaying the results as both a table and a list (permalink):
Semiprime | Partitions ----------------------- 4 | 1 6 | 1 9 | 1 10 | 2 14 | 3 15 | 2 21 | 4 22 | 7 25 | 7 26 | 10 33 | 17 34 | 25 35 | 21 38 | 34 39 | 34 46 | 73 49 | 87 51 | 103 55 | 149 57 | 176 58 | 206 62 | 281 65 | 344 69 | 479 74 | 725 77 | 881 82 | 1311 85 | 1597 86 | 1742 87 | 1841 91 | 2445 93 | 2808 94 | 3052 95 | 3222 106 | 6784 111 | 9298 115 | 11989 118 | 14533 119 | 15384 121 | 17414 122 | 18581 123 | 19680 129 | 28284 133 | 35862 134 | 38125 141 | 57095 142 | 60582 143 | 64010 145 | 71730 146 | 76016
1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016
As can be seen, the integer \( \textbf{43} \) has \( \textbf{28284} \) partitions compared to its 63261 unrestricted partitions.
Similar results could be created for primes and triprimes. Here are the result for the primes (permalink). The numbers form OEIS A056768.
Prime | Partitions --------------------- 2 | 1 3 | 1 5 | 2 7 | 3 11 | 6 13 | 9 17 | 17 19 | 23 23 | 40 29 | 87 31 | 111 37 | 219 41 | 336 43 | 413 47 | 614 53 | 1083 59 | 1850 61 | 2198 67 | 3630 71 | 5007 73 | 5861 79 | 9282 83 | 12488 89 | 19232 97 | 33439 101 | 43709 103 | 49871 107 | 64671 109 | 73506 113 | 94625 127 | 221265 131 | 279516 137 | 394170 139 | 441250 149 | 766262 151 | 853692 157 | 1175344 163 | 1608014 167 | 1975108 173 | 2675925 179 | 3605666 181 | 3977861 191 | 6447003 193 | 7089299 197 | 8559069 199 | 9397474 211 | 16298212 223 | 27810910 227 | 33121140 229 | 36123177
1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925, 3605666, 3977861, 6447003, 7089299, 8559069, 9397474, 16298212, 27810910, 33121140, 36123177
Here are the result for the triprimes (permalink).
Triprime | Partitions ----------------------- 8 | 1 12 | 1 18 | 1 20 | 2 27 | 1 28 | 3 30 | 2 42 | 4 44 | 7 45 | 2 50 | 7 52 | 10 63 | 7 66 | 18 68 | 25 70 | 22 75 | 13 76 | 34 78 | 36 92 | 78 98 | 97 99 | 47 102 | 117 105 | 60 110 | 170 114 | 202 116 | 230 117 | 111 124 | 316 125 | 165 130 | 401 138 | 579 147 | 478 148 | 866 153 | 616 154 | 1086 164 | 1652 165 | 1040 170 | 2076 171 | 1350 172 | 2227 174 | 2418 175 | 1566 182 | 3271 186 | 3802 188 | 4115 190 | 4372 195 | 3587 207 | 5708 212 | 9783
1, 1, 1, 2, 1, 3, 2, 4, 7, 2, 7, 10, 7, 18, 25, 22, 13, 34, 36, 78, 97, 47, 117, 60, 170, 202, 230, 111, 316, 165, 401, 579, 478, 866, 616, 1086, 1652, 1040, 2076, 1350, 2227, 2418, 1566, 3271, 3802, 4115, 4372, 3587, 5708, 9783, 13878, 18291, 13983, 22465, 23913, 27489, 29296, 23092, 31441, 32846 (these are first 60 terms).
These results are base-independent and in general what is being done here is to take an integer with a certain property (primeness for example) and express it as a sum of smaller numbers with the same property. The numbere of ways in which this can be done is being counted. Here are some examples:
- 129 = 4 + 4 + 4 + 6 + 111 ... there are 28284 ways in which this can be done
- 97 = 3 + 5 + 89 ... there are 33439 ways in which this can be done
- 212 = 8 + 12 + 18 + 174 ... there are 9783 ways in which this can be done
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