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Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.
22

%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