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.

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Author

Gus Wiseman, Aug 01 2023

Keywords

• nonn

Comments

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

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

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.

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

Examples

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}
		

Crossrefs

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

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

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

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

The complement is A364531.

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

A000005 counts divisors, nonprime A033273, composite A055212.

A001222 counts prime indices.

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

A112798 lists prime indices, sum A056239.

A299701 counts distinct subset-sums of prime indices.

A299702 ranks knapsack partitions, complement A299729.

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

Programs

Mathematica

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