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A066417
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Sum of anti-divisors of n.
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34
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0, 0, 2, 3, 5, 4, 10, 8, 8, 14, 12, 13, 19, 16, 18, 14, 28, 28, 18, 24, 22, 36, 34, 23, 39, 24, 42, 46, 24, 36, 42, 58, 48, 30, 52, 32, 50, 70, 52, 55, 41, 66, 56, 40, 86, 58, 60, 56, 72, 80, 42, 94, 88, 52, 74, 56, 74, 96, 90, 107, 57, 78, 112, 46, 84, 86, 132, 112, 54, 102
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| See A066272 for definition of anti-divisor.
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LINKS
| Jon Perry, Anti-divisors [Broken link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]
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FORMULA
| G.f. sum(k>0, 2k x^(3k) / (1 - x^(2k)) + (2k+1)(x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1))). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
For n>1, a(n) = A000203(2*n-1) + A000203(2*n+1) + A000203(n/2^k)*2^(k+1) - 6*n - 2, where k=A007814(n). [From Max Alekseyev (maxale(AT)gmail.com), Apr 27 2010]
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EXAMPLE
| For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 28.
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MAPLE
| antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:
A066417 := proc(n) add(d, d=antidivisors(n)) ; end proc: # R. J. Mathar, Jul 04 2011
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MATHEMATICA
| antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n &]; Table[ Plus @@ antid[n], {n, 70}] (from Robert G. Wilson v Mar 15 2004)
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PROG
| (PARI) al(n)=Vec(sum(k=1, n, 2*k*(x^(3*k)+x*O(x^n))/(1-x^(2*k))+(2*k+1)*(x^(3*k+1)+x^(3*k+2)+x*O(x^n))/(1-x^(2*k+1)))) [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
(PARI) { a(n) = local(k); if(n>1, k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0) } [From Max Alekseyev (maxale(AT)gmail.com), Apr 27 2010]
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CROSSREFS
| Cf. A066416, A066418, A058838, A064277.
Sequence in context: A100932 A064360 A075158 * A079521 A112060 A084933
Adjacent sequences: A066414 A066415 A066416 * A066418 A066419 A066420
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KEYWORD
| nonn
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AUTHOR
| Jon Perry (perry(AT)globalnet.co.uk), Dec 28 2001
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