cp's OEIS Frontend

A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

%I A364348 #8 Oct 18 2023 04:44:56
%S A364348 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,105,
%T A364348 108,114,120,126,130,132,133,138,140,144,147,150,154,156,162,165,168,
%U A364348 174,180,186,189,192,195,198,204,210,216,222,228,231,234,240,246,252
%N A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.
%C A364348 Or numbers with a prime index equal to the sum of two others, allowing re-used parts.
%C A364348 Also Heinz numbers of a type of sum-free partitions counted by A363225.
%e A364348 We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.
%e A364348 The terms together with their prime indices begin:
%e A364348    6: {1,2}
%e A364348   12: {1,1,2}
%e A364348   18: {1,2,2}
%e A364348   21: {2,4}
%e A364348   24: {1,1,1,2}
%e A364348   30: {1,2,3}
%e A364348   36: {1,1,2,2}
%e A364348   42: {1,2,4}
%e A364348   48: {1,1,1,1,2}
%e A364348   54: {1,2,2,2}
%e A364348   60: {1,1,2,3}
%e A364348   63: {2,2,4}
%e A364348   65: {3,6}
%e A364348   66: {1,2,5}
%e A364348   70: {1,3,4}
%e A364348   72: {1,1,1,2,2}
%t A364348 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A364348 Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]!={}&]
%Y A364348 Subsets of this type are counted by A093971, complement A007865.
%Y A364348 Partitions of this type are counted by A363225, strict A363226.
%Y A364348 The complement is A364347, counted by A364345.
%Y A364348 The complement without re-using parts is A364461, counted by A236912.
%Y A364348 Without re-using parts we have A364462, counted by A237113.
%Y A364348 A001222 counts prime indices.
%Y A364348 A108917 counts knapsack partitions, ranks A299702.
%Y A364348 A112798 lists prime indices, sum A056239.
%Y A364348 A323092 counts double-free partitions, ranks A320340.
%Y A364348 Cf. A237667, A275972, A288728, A325862, A326083, A364346.
%K A364348 nonn
%O A364348 1,1
%A A364348 _Gus Wiseman_, Jul 27 2023