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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-François 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.)