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Search: id:A066417
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| A066417 |
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Sum of anti-divisors of n. |
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+0
22
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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; listen)
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OFFSET
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1,3
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COMMENT
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See A066272 for definition of anti-divisor.
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LINKS
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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
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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]
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EXAMPLE
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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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MATHEMATICA
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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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PROGRAM
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(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]
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CROSSREFS
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Cf. A066416, A066418, A058838, A064277.
Sequence in context: A100932 A064360 A075158 this_sequence A079521 A112060 A084933
Adjacent sequences: A066414 A066415 A066416 this_sequence A066418 A066419 A066420
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KEYWORD
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nonn
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AUTHOR
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Jon Perry (perry(AT)globalnet.co.uk), Dec 28 2001
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