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A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

%I A364462 #11 Aug 30 2023 14:59:46
%S A364462 12,24,30,36,48,60,63,70,72,84,90,96,108,120,126,132,140,144,150,154,
%T A364462 156,165,168,180,189,192,204,210,216,228,240,252,264,270,273,276,280,
%U A364462 286,288,300,308,312,315,324,325,330,336,348,350,360,372,378,384,390
%N A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.
%C A364462 Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.
%C A364462 No partitions of this type are knapsack (A299702, A299729).
%C A364462 All multiples of terms are terms. - _Robert Israel_, Aug 30 2023
%H A364462 Robert Israel, <a href="/A364462/b364462.txt">Table of n, a(n) for n = 1..10000</a>
%e A364462 The terms together with their prime indices begin:
%e A364462    12: {1,1,2}
%e A364462    24: {1,1,1,2}
%e A364462    30: {1,2,3}
%e A364462    36: {1,1,2,2}
%e A364462    48: {1,1,1,1,2}
%e A364462    60: {1,1,2,3}
%e A364462    63: {2,2,4}
%e A364462    70: {1,3,4}
%e A364462    72: {1,1,1,2,2}
%e A364462    84: {1,1,2,4}
%e A364462    90: {1,2,2,3}
%e A364462    96: {1,1,1,1,1,2}
%e A364462   108: {1,1,2,2,2}
%e A364462   120: {1,1,1,2,3}
%e A364462   126: {1,2,2,4}
%e A364462   132: {1,1,2,5}
%e A364462   140: {1,1,3,4}
%e A364462   144: {1,1,1,1,2,2}
%p A364462 filter:= proc(n) local F, i,j,m;
%p A364462   F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
%p A364462   for i from 1 to nops(F)-1 do for j from 1 to i-1 do
%p A364462     if member(F[i]+F[j],F) then return true fi
%p A364462   od od;
%p A364462   false
%p A364462 end proc:
%p A364462 select(filter, [$1..1000]); # _Robert Israel_, Aug 30 2023
%t A364462 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A364462 Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]!={}&]
%Y A364462 Subsets not of this type are counted by A085489, w/ re-usable parts A007865.
%Y A364462 Subsets of this type are counted by A088809, with re-usable parts A093971.
%Y A364462 Partitions not of this type are counted by A236912.
%Y A364462 Partitions of this type are counted by A237113.
%Y A364462 Subset of A299729.
%Y A364462 The complement with re-usable parts is A364347, counted by A364345.
%Y A364462 With re-usable parts we have A364348, counted by A363225 (strict A363226).
%Y A364462 The complement is A364461.
%Y A364462 The non-binary complement is A364531, counted by A237667.
%Y A364462 The non-binary version is A364532, see also A364350.
%Y A364462 A001222 counts prime indices.
%Y A364462 A108917 counts knapsack partitions, ranks A299702.
%Y A364462 A112798 lists prime indices, sum A056239.
%Y A364462 Cf. A151897, A320340, A325862, A326083.
%K A364462 nonn
%O A364462 1,1
%A A364462 _Gus Wiseman_, Jul 29 2023