A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390

Offset: 1

Author

Gus Wiseman, Jul 29 2023

Keywords

• nonn

Comments

Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.

No partitions of this type are knapsack (A299702, A299729).

All multiples of terms are terms. - Robert Israel, Aug 30 2023

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}
   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}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsets not of this type are counted by A085489, w/ re-usable parts A007865.

Subsets of this type are counted by A088809, with re-usable parts A093971.

Partitions not of this type are counted by A236912.

Partitions of this type are counted by A237113.

Subset of A299729.

The complement with re-usable parts is A364347, counted by A364345.

With re-usable parts we have A364348, counted by A363225 (strict A363226).

The complement is A364461.

The non-binary complement is A364531, counted by A237667.

The non-binary version is A364532, see also A364350.

A001222 counts prime indices.

A108917 counts knapsack partitions, ranks A299702.

A112798 lists prime indices, sum A056239.

Cf. A151897, A320340, A325862, A326083.

Programs

Maple

filter:= proc(n) local F, i,j,m;
  F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
  for i from 1 to nops(F)-1 do for j from 1 to i-1 do
    if member(F[i]+F[j],F) then return true fi
  od od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023

Mathematica

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