%I #15 Oct 18 2023 04:44:40
%S 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,
%T 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U 59,61,62,64,67,68,69,71,73,74,75,76,77,79,80,81,82,83,85
%N Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.
%C Or numbers without any prime index equal to the sum of two others, allowing re-used parts.
%C Also Heinz numbers of a type of sum-free partitions counted by A364345.
%e We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
%e The terms together with their prime indices begin:
%e 1: {}
%e 2: {1}
%e 3: {2}
%e 4: {1,1}
%e 5: {3}
%e 7: {4}
%e 8: {1,1,1}
%e 9: {2,2}
%e 10: {1,3}
%e 11: {5}
%e 13: {6}
%e 14: {1,4}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 17: {7}
%e 19: {8}
%e 20: {1,1,3}
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]=={}&]
%Y Subsets of this type are counted by A007865 (sum-free sets).
%Y Partitions of this type are counted by A364345.
%Y The squarefree case is counted by A364346.
%Y The complement is A364348, counted by A363225.
%Y The non-binary version is counted by A364350.
%Y Without re-using parts we have A364461, counted by A236912.
%Y Without re-using parts we have complement A364462, counted by A237113.
%Y A001222 counts prime indices.
%Y A108917 counts knapsack partitions, ranks A299702.
%Y A112798 lists prime indices, sum A056239.
%Y A323092 counts double-free partitions, ranks A320340.
%Y Cf. A093971, A237667, A288728, A325862, A326083, A363226, A364531.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 26 2023