A065205 Number of subsets of proper divisors of n that sum to n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 34, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 1, 0, 6, 0, 0, 0, 25, 0, 0, 0, 1, 0, 23, 0, 0, 0, 0, 0, 21, 0, 0, 0, 2

Offset

1,12

Comments

Deficient and weird numbers have a(n) = 0, perfect numbers and others (see A064771) have a(n) = 1.

Number of partitions of n into distinct proper divisors of n; a(A136447(n)) = 0; a(A005835(n)) > 0; a(A064771(n)) = 1. - Reinhard Zumkeller, Jan 21 2013

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Formula

a(n) = A033630(n) - 1.

Example

a(20) = 1 because {1, 4, 5, 10} is the only subset of proper divisors of 20 that sum to 20.
a(24) = 5 because there are five different subsets we can use to sum up to 24: {1, 2, 3, 4, 6, 8}, {1, 2, 3, 6, 12}, {1, 3, 8, 12}, {2, 4, 6, 12}, {4, 8, 12}.

Mathematica

a[n_] := (dd = Most[ Divisors[n] ]; cc = Array[c, Length[dd]]; Length[ {ToRules[ Reduce[ And @@ (0 <= # <= 1 &) /@ cc && dd . cc == n, cc, Integers]]}]); Table[ a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 23 2012 *)

Prog

(Haskell)
a065205 n = p (a027751_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jan 21 2013
(PARI) a(n, s, d)={s || (s=sigma(n)-n) || return; d||d=vecextract(divisors(n), "^-1"); while(d[#d]>n, s-=d[#d]; d=d[1..-2]); s<=n && return(s==n); if( n>d[#d], a(n-d[#d], s-d[#d], d[1..-2]), 1)+a(n, s-d[#d], d[1..-2])} \\ M. F. Hasler, May 11 2015

Crossrefs

Cf. A064771, A005835.

Cf. A065218 for records.

Cf. A027751, A210442, A211110, A033630.

Sequence in context: A341624 A341620 A097798 * A036272 A257403 A326559

Adjacent sequences: A065202 A065203 A065204 * A065206 A065207 A065208

Keyword

nonn

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001

Extensions

More terms and additional comments from Jud McCranie, Oct 21 2001

Status

approved