A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Author

Gus Wiseman, Jul 26 2023

Keywords

• nonn

Comments

Or numbers without any 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 A364345.

Examples

			We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
		

Crossrefs

Subsets of this type are counted by A007865 (sum-free sets).

Partitions of this type are counted by A364345.

The squarefree case is counted by A364346.

The complement is A364348, counted by A363225.

The non-binary version is counted by A364350.

Without re-using parts we have A364461, counted by A236912.

Without re-using parts we have complement 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. A093971, A237667, A288728, A325862, A326083, A363226, A364531.

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]]=={}&]