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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.
19
12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390
OFFSET
1,1
COMMENTS
Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.
No partitions of this type are knapsack (A299702, A299729).
All multiples of terms are terms. - Robert Israel, Aug 30 2023
LINKS
EXAMPLE
The terms together with their prime indices begin:
12: {1,1,2}
24: {1,1,1,2}
30: {1,2,3}
36: {1,1,2,2}
48: {1,1,1,1,2}
60: {1,1,2,3}
63: {2,2,4}
70: {1,3,4}
72: {1,1,1,2,2}
84: {1,1,2,4}
90: {1,2,2,3}
96: {1,1,1,1,1,2}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
MAPLE
filter:= proc(n) local F, i, j, m;
F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
for i from 1 to nops(F)-1 do for j from 1 to i-1 do
if member(F[i]+F[j], F) then return true fi
od od;
false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Intersection[prix[#], Total/@Subsets[prix[#], {2}]]!={}&]
CROSSREFS
Subsets not of this type are counted by A085489, w/ re-usable parts A007865.
Subsets of this type are counted by A088809, with re-usable parts A093971.
Partitions not of this type are counted by A236912.
Partitions of this type are counted by A237113.
Subset of A299729.
The complement with re-usable parts is A364347, counted by A364345.
With re-usable parts we have A364348, counted by A363225 (strict A363226).
The complement is A364461.
The non-binary complement is A364531, counted by A237667.
The non-binary version is A364532, see also A364350.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Sequence in context: A350056 A334799 A352287 * A369182 A108938 A085236
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2023
STATUS
approved