A299702 - OEIS

A299702

Heinz numbers of knapsack partitions.

111

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, 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, 77, 78

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OFFSET

1,2

COMMENTS

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

LINKS

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

MAPLE

filter:= proc(n) local F, t, S, i, r;

F:= map(t -> [numtheory:-pi(t[1]), t[2]], ifactors(n)[2]);

S:= {0}: r:= 1;

for t in F do

S:= map(s -> seq(s + i*t[1], i=0..t[2]), S);

r:= r*(t[2]+1);

if nops(S) <> r then return false fi

od;

true

end proc:

select(filter, [$1..100]); # Robert Israel, Oct 30 2024

MATHEMATICA

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], UnsameQ@@Plus@@@Union[Rest@Subsets[primeMS[#]]]&]

CROSSREFS

Cf. A056239, A108917, A112798, A275972, A276024, A284640, A296150, A299701, A299729.

Sequence in context: A160453 A325778 A364531 * A348577 A361232 A027855

Adjacent sequences: A299699 A299700 A299701 * A299703 A299704 A299705

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 17 2018

STATUS

approved