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A083206
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Number of subsets of divisors of n having equal sums as their complements.
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10
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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; internal format)
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OFFSET
| 1,24
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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 (reinhard.zumkeller(AT)gmail.com), Jul 10 2010]
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LINKS
| T. D. Noe and R. Zumkeller, Table of n, a(n) for n=1..10000
Reinhard Zumkeller, Illustration of initial terms
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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 (reinhard.zumkeller(AT)gmail.com), Jul 10 2010]
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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.
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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
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KEYWORD
| nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2003
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