A364532 - OEIS

%I #10 Aug 03 2023 09:04:25

%S 12,24,30,36,40,48,60,63,70,72,80,84,90,96,108,112,120,126,132,140,

%T 144,150,154,156,160,165,168,180,189,192,198,200,204,210,216,220,224,

%U 228,240,252,264,270,273,276,280,286,288,300,308,312,315,320,324,325,330

%N 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 First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.

%C First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.

%C 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 These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

%e The terms together with their prime indices begin:

%e   12: {1,1,2}

%e   24: {1,1,1,2}

%e   30: {1,2,3}

%e   36: {1,1,2,2}

%e   40: {1,1,1,3}

%e   48: {1,1,1,1,2}

%e   60: {1,1,2,3}

%e   63: {2,2,4}

%e   70: {1,3,4}

%e   72: {1,1,1,2,2}

%e   80: {1,1,1,1,3}

%e   84: {1,1,2,4}

%e   90: {1,2,2,3}

%e   96: {1,1,1,1,1,2}

%t Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

%Y Partitions not of this type are counted by A237667, strict A364349.

%Y Partitions of this type are counted by A237668, strict A364272.

%Y The binary complement is A364461, re-usable A364347 (counted by A364345).

%Y The binary version is A364462, re-usable A364348 (counted by A363225).

%Y The complement is A364531.

%Y Subsets of this type are counted by A364534, complement A151897.

%Y A000005 counts divisors, nonprime A033273, composite A055212.

%Y A001222 counts prime indices.

%Y A108917 counts knapsack partitions, strict A275972, for subsets A325864.

%Y A112798 lists prime indices, sum A056239.

%Y A299701 counts distinct subset-sums of prime indices.

%Y A299702 ranks knapsack partitions, complement A299729.

%Y Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 01 2023