OFFSET
1,24
COMMENTS
a(n)=0 for deficient numbers n (A005100), but the converse is not true, as 18 is abundant (A005101) and a(18)=0, see A083211;
In order that a(n)>0, the sum of divisors of n must be even by definition: a(n) = half the number of partitions of A000203(n)/2 into divisors of n, see formula. [Reinhard Zumkeller, Jul 10 2010]
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000 (terms 1..800 from Reinhard Zumkeller).
Reinhard Zumkeller, Illustration of initial terms
FORMULA
a(n) = if sigma(n) mod 2 = 1 then 0 else f(n,sigma(n)/2,2), where sigma=A000203 and f(n,m,k) = if k<=m then f(n,m,k+1)+f(n,m-k,k+1)*0^(n mod k) else 0^m, cf. A033630, also using f. [Reinhard Zumkeller, Jul 10 2010]
a(n) is half the coefficient of x^0 in Product_{d|n} (x^d + 1/x^d). - Ilya Gutkovskiy, Feb 04 2024
EXAMPLE
a(24)=3: 1+2+3+4+8+12=6+24, 1+3+6+8+12=2+4+24, 4+6+8+12=1+2+3+24.
MATHEMATICA
a[n_] := (s = DivisorSigma[1, n]; If[Mod[s, 2] == 1, 0, f[n, s/2, 2]]); f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k+1] + f[n, m-k, k+1]*Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[a, 105] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 22 2003
STATUS
approved