A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252

Offset: 1

Author

Gus Wiseman, Jul 27 2023

Keywords

• nonn

Comments

Or numbers with a prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A363225.

Examples

			We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.
The terms together with their prime indices begin:
   6: {1,2}
  12: {1,1,2}
  18: {1,2,2}
  21: {2,4}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  70: {1,3,4}
  72: {1,1,1,2,2}
		

Crossrefs

Subsets of this type are counted by A093971, complement A007865.

Partitions of this type are counted by A363225, strict A363226.

The complement is A364347, counted by A364345.

The complement without re-using parts is A364461, counted by A236912.

Without re-using parts we have A364462, counted by A237113.

A001222 counts prime indices.

A108917 counts knapsack partitions, ranks A299702.

A112798 lists prime indices, sum A056239.

A323092 counts double-free partitions, ranks A320340.

Cf. A237667, A275972, A288728, A325862, A326083, A364346.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]!={}&]