A305611 Number of distinct positive subset-sums of the multiset of prime factors of n.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 9, 1, 3, 5, 6, 3, 7, 1, 5, 3, 6, 1, 10, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 10, 3, 3

Offset: 1

Author

Gus Wiseman, Jun 06 2018

Keywords

• nonn

Comments

An integer n is a positive subset-sum of a multiset y if there exists a nonempty submultiset of y with sum n.

One less than the number of distinct values obtained when A001414 is applied to all divisors of n. - Antti Karttunen, Jun 13 2018

Examples

			The a(12) = 5 positive subset-sums of {2, 2, 3} are 2, 3, 4, 5, and 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A001414, A027746, A056239, A122768, A276024, A284640, A299701, A299702, A301855, A301935, A301957, A304792, A304793, A305613.

Programs

Mathematica

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[p,{k}]]]]]],{n,100}]

PARI

up_to = 65537;
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
v001414 = vector(up_to,n,A001414(n));
A305611(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v001414[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jun 13 2018

Python

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A305611(n):
    fs = factorint(n)
    return len(set(sum(d) for i in range(1,sum(fs.values())+1) for d in multiset_combinations(fs,i))) # Chai Wah Wu, Aug 23 2021