A364461 - OEIS

A364461

Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

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OFFSET

1,2

COMMENTS

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

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, Jan 27 2026

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 of this type are counted by A085489, with re-usable parts A007865.

Subsets not of this type are counted by A093971, w/ re-usable parts A088809.

Partitions of this type are counted by A236912.

Allowing parts to be re-used gives A364347, counted by A364345.

The complement allowing parts to be re-used is A364348, counted by A363225.

The non-binary version allowing re-used parts is counted by A364350.

The complement is A364462, counted by A237113.

The non-binary version is A364531, counted by A237667, complement A364532.

A001222 counts prime indices.

A108917 counts knapsack partitions, ranks A299702.

A112798 lists prime indices, sum A056239.

Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

Sequence in context: A168186 A353866 A085235 * A160453 A325778 A364531

Adjacent sequences: A364458 A364459 A364460 * A364462 A364463 A364464

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 27 2023

STATUS

approved