A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 14, 0, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

• nonn

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;

a(n)=1 for perfect numbers n (A000396), see A083209 for all numbers with a(n)=1;

records: A083213(k)=a(A083212(k)).

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]

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..101632 (terms 1..800 from Reinhard Zumkeller, terms 801..10000 from T. D. Noe).
• Reinhard Zumkeller, Illustration of initial terms

Crossrefs

Cf. A083208 [= a(A083207(n))], A083211, A000005, A000203, A082729, A378446 (inverse Möbius transform), A378449.

Cf. A083207 (positions of terms > 0), A083210 (positions of 0's), A083209 (positions of 1's), A378652 (of 2's).

Cf. also A033630, A065205, A058377, A325806.

Programs

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 *)

PARI

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); }; \\ Antti Karttunen, Dec 02 2024, after Ilya Gutkovskiy

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