cp's OEIS Frontend

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

%I A364532 #10 Aug 03 2023 09:04:25
%S A364532 12,24,30,36,40,48,60,63,70,72,80,84,90,96,108,112,120,126,132,140,
%T A364532 144,150,154,156,160,165,168,180,189,192,198,200,204,210,216,220,224,
%U A364532 228,240,252,264,270,273,276,280,286,288,300,308,312,315,320,324,325,330
%N A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.
%C A364532 First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
%C A364532 First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
%C A364532 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A364532 These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.
%e A364532 The terms together with their prime indices begin:
%e A364532   12: {1,1,2}
%e A364532   24: {1,1,1,2}
%e A364532   30: {1,2,3}
%e A364532   36: {1,1,2,2}
%e A364532   40: {1,1,1,3}
%e A364532   48: {1,1,1,1,2}
%e A364532   60: {1,1,2,3}
%e A364532   63: {2,2,4}
%e A364532   70: {1,3,4}
%e A364532   72: {1,1,1,2,2}
%e A364532   80: {1,1,1,1,3}
%e A364532   84: {1,1,2,4}
%e A364532   90: {1,2,2,3}
%e A364532   96: {1,1,1,1,1,2}
%t A364532 Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]
%Y A364532 Partitions not of this type are counted by A237667, strict A364349.
%Y A364532 Partitions of this type are counted by A237668, strict A364272.
%Y A364532 The binary complement is A364461, re-usable A364347 (counted by A364345).
%Y A364532 The binary version is A364462, re-usable A364348 (counted by A363225).
%Y A364532 The complement is A364531.
%Y A364532 Subsets of this type are counted by A364534, complement A151897.
%Y A364532 A000005 counts divisors, nonprime A033273, composite A055212.
%Y A364532 A001222 counts prime indices.
%Y A364532 A108917 counts knapsack partitions, strict A275972, for subsets A325864.
%Y A364532 A112798 lists prime indices, sum A056239.
%Y A364532 A299701 counts distinct subset-sums of prime indices.
%Y A364532 A299702 ranks knapsack partitions, complement A299729.
%Y A364532 Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.
%K A364532 nonn
%O A364532 1,1
%A A364532 _Gus Wiseman_, Aug 01 2023