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

1,4

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

Links

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A239707 A294928 A325770 * A325765 A032741 A364818

Adjacent sequences: A305608 A305609 A305610 * A305612 A305613 A305614

Keyword

nonn

Author

Gus Wiseman, Jun 06 2018

Status

approved