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Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.
16

%I #8 Oct 18 2023 04:44:56

%S 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,105,

%T 108,114,120,126,130,132,133,138,140,144,147,150,154,156,162,165,168,

%U 174,180,186,189,192,195,198,204,210,216,222,228,231,234,240,246,252

%N Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

%C Or numbers with a 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 A363225.

%e We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.

%e The terms together with their prime indices begin:

%e 6: {1,2}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 21: {2,4}

%e 24: {1,1,1,2}

%e 30: {1,2,3}

%e 36: {1,1,2,2}

%e 42: {1,2,4}

%e 48: {1,1,1,1,2}

%e 54: {1,2,2,2}

%e 60: {1,1,2,3}

%e 63: {2,2,4}

%e 65: {3,6}

%e 66: {1,2,5}

%e 70: {1,3,4}

%e 72: {1,1,1,2,2}

%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 A093971, complement A007865.

%Y Partitions of this type are counted by A363225, strict A363226.

%Y The complement is A364347, counted by A364345.

%Y The complement without re-using parts is A364461, counted by A236912.

%Y Without re-using parts we have 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. A237667, A275972, A288728, A325862, A326083, A364346.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 27 2023