cp's OEIS Frontend

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

%I A364347 #15 Oct 18 2023 04:44:40
%S A364347 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 A364347 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U A364347 59,61,62,64,67,68,69,71,73,74,75,76,77,79,80,81,82,83,85
%N A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.
%C A364347 Or numbers without any prime index equal to the sum of two others, allowing re-used parts.
%C A364347 Also Heinz numbers of a type of sum-free partitions counted by A364345.
%e A364347 We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
%e A364347 The terms together with their prime indices begin:
%e A364347     1: {}
%e A364347     2: {1}
%e A364347     3: {2}
%e A364347     4: {1,1}
%e A364347     5: {3}
%e A364347     7: {4}
%e A364347     8: {1,1,1}
%e A364347     9: {2,2}
%e A364347    10: {1,3}
%e A364347    11: {5}
%e A364347    13: {6}
%e A364347    14: {1,4}
%e A364347    15: {2,3}
%e A364347    16: {1,1,1,1}
%e A364347    17: {7}
%e A364347    19: {8}
%e A364347    20: {1,1,3}
%t A364347 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A364347 Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]=={}&]
%Y A364347 Subsets of this type are counted by A007865 (sum-free sets).
%Y A364347 Partitions of this type are counted by A364345.
%Y A364347 The squarefree case is counted by A364346.
%Y A364347 The complement is A364348, counted by A363225.
%Y A364347 The non-binary version is counted by A364350.
%Y A364347 Without re-using parts we have A364461, counted by A236912.
%Y A364347 Without re-using parts we have complement A364462, counted by A237113.
%Y A364347 A001222 counts prime indices.
%Y A364347 A108917 counts knapsack partitions, ranks A299702.
%Y A364347 A112798 lists prime indices, sum A056239.
%Y A364347 A323092 counts double-free partitions, ranks A320340.
%Y A364347 Cf. A093971, A237667, A288728, A325862, A326083, A363226, A364531.
%K A364347 nonn
%O A364347 1,2
%A A364347 _Gus Wiseman_, Jul 26 2023