A299702 Heinz numbers of knapsack partitions.

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

Offset: 1

Author

Gus Wiseman, Feb 17 2018

Keywords

• nonn

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

Crossrefs

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

Programs

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[#]]]&]