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

%I A375154 #22 Aug 06 2024 21:35:18
%S A375154 5,25,33,39,38,58,65,86,123,85,82,92,98,152,99,158,106,135,153,145,
%T A375154 215,186,142,189,235,178,185,165,147,315,274,231,214,305,171,332,207,
%U A375154 290,310,344,266,358,583,297,261,278,285,488,255,334,302,369,309,2888,284
%N A375154 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z).
%C A375154 No other prime apart from initial 5.
%H A375154 David A. Corneth, <a href="/A375154/b375154.txt">Table of n, a(n) for n = 1..764</a>
%e A375154 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:
%e A375154 sigma(2) + sigma(14) + sigma(49) = 3 + 24 + 57 = 84;
%e A375154 sigma(3) + sigma(7) + sigma(55) = 4 + 8 + 72 = 84;
%e A375154 sigma(3) + sigma(23) + sigma(39) = 4 + 24 + 56 = 84;
%e A375154 sigma(5) + sigma(5) + sigma(55) = 6 + 6 + 72 = 84;
%e A375154 sigma(7) + sigma(19) + sigma(39) = 8 + 20 + 56 = 84;
%e A375154 sigma(10) + sigma(14) + sigma(41) = 18 + 24 + 42 = 84;
%e A375154 sigma(13) + sigma(13) + sigma(39) = 14 + 14 + 56 = 84;
%e A375154 Furthermore 65 is the minimum number to have this property.
%t A375154 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 *)
%Y A375154 Cf. A000203, A211223, A211224, A211225, A373047.
%K A375154 nonn
%O A375154 1,1
%A A375154 _Paolo P. Lava_, Aug 01 2024