A373047 -id:A373047 - cp's OEIS frontend

A375154 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z).

5, 25, 33, 39, 38, 58, 65, 86, 123, 85, 82, 92, 98, 152, 99, 158, 106, 135, 153, 145, 215, 186, 142, 189, 235, 178, 185, 165, 147, 315, 274, 231, 214, 305, 171, 332, 207, 290, 310, 344, 266, 358, 583, 297, 261, 278, 285, 488, 255, 334, 302, 369, 309, 2888, 284

Offset: 1

Paolo P. Lava, Aug 01 2024

nonn

No other prime apart from initial 5.

			a(7) = 65 and 65 has 7 partitions of three numbers, x, y and z, for which sigma(65) = sigma(x) + sigma(y) + sigma(z) = 84. In fact:
sigma(2) + sigma(14) + sigma(49) = 3 + 24 + 57 = 84;
sigma(3) + sigma(7) + sigma(55) = 4 + 8 + 72 = 84;
sigma(3) + sigma(23) + sigma(39) = 4 + 24 + 56 = 84;
sigma(5) + sigma(5) + sigma(55) = 6 + 6 + 72 = 84;
sigma(7) + sigma(19) + sigma(39) = 8 + 20 + 56 = 84;
sigma(10) + sigma(14) + sigma(41) = 18 + 24 + 42 = 84;
sigma(13) + sigma(13) + sigma(39) = 14 + 14 + 56 = 84;
Furthermore 65 is the minimum number to have this property.

David A. Corneth, Table of n, a(n) for n = 1..764

Cf. A000203, A211223, A211224, A211225, A373047.

f[n_] := Count[IntegerPartitions[n, {3}], ?(Total[DivisorSigma[1, #]] == DivisorSigma[1, n] &)]; With[{s = Array[f, 600]}, TakeWhile[FirstPosition[s, #] & /@ Range[Max[s]] // Flatten, !MissingQ[#] &]] (* _Amiram Eldar, Aug 02 2024 *)

A375238 Least k with exactly n partitions k = x + y + z satisfying k' = x' + y' + z', where k' is the arithmetic derivative of k.

Original entry on oeis.org

5, 9, 22, 35, 65, 63, 70, 62, 82, 110, 75, 143, 130, 169, 142, 186, 170, 194, 230, 284, 234, 195, 147, 345, 238, 245, 323, 290, 286, 294, 285, 334, 430, 534, 458, 255, 385, 434, 390, 418, 374, 399, 441, 526, 518, 382, 748, 598, 578, 454, 455, 585, 507, 435, 582

Offset: 1

Paolo P. Lava, Aug 06 2024

			a(7) = 70 and 70 has 7 partitions of three numbers, x, y and z, for which 70' = x' + y' + z' = 59. In fact:
5' + 21' + 44' = 1 + 10 + 48 = 59;
6' + 14' + 50' = 5 + 9 + 45 = 59;
6' + 22' + 42' = 5 + 13 + 41 = 59;
10' + 10' + 50' = 7 + 7 + 45 = 59;
13' + 24' + 33' = 1 + 44 + 14 = 59;
13' + 27' + 30' = 1 + 27 + 31 = 59;
14' + 14' + 42' = 9 + 9 + 41 = 59.
Furthermore 70 is the minimum number to have this property.

Cf. A003415, A212662, A373047, A375154.

cp's OEIS Frontend

A375154 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z).

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