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A083206 Number of subsets of divisors of n having equal sums as their complements. 12
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 (list; graph; refs; listen; history; text; internal format)
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;

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. [From Reinhard Zumkeller, Jul 10 2010]

LINKS

T. D. Noe and R. Zumkeller, Table of n, a(n) for n=1..10000

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. [From Reinhard Zumkeller, Jul 10 2010]

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-Fran├žois Alcover, Jul 29 2015, after Reinhard Zumkeller *)

CROSSREFS

Cf. A083207, A083208, A083209, A083210, A083211, A000005, A000203, A082729, A033630, A065205.

Sequence in context: A128980 A096693 A193139 * A069531 A035677 A143276

Adjacent sequences:  A083203 A083204 A083205 * A083207 A083208 A083209

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Apr 22 2003

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)