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Search: id:A002129
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| A002129 |
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Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.
(Formerly M3236 N1307)
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+0
20
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| 1, -1, 4, -5, 6, -4, 8, -13, 13, -6, 12, -20, 14, -8, 24, -29, 18, -13, 20, -30, 32, -12, 24, -52, 31, -14, 40, -40, 30, -24, 32, -61, 48, -18, 48, -65, 38, -20, 56, -78, 42, -32, 44, -60, 78, -24, 48, -116, 57, -31, 72, -70, 54, -40, 72, -104, 80, -30, 60, -120, 62, -32, 104, -125
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n)^2 = Sum_{n >= 1} (-1)^(n-1)*n*x^n/(1-x^n).
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 3rd formula.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. R. Finch, The "One-Ninth" Constant
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FORMULA
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Multiplicative with a(p^e) = 3-2^(e+1) if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Sep 01, 2001
G.f.: Sum_{n>=1} n*x^n*(1-3*x^n)/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 15 2002
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log[ Sum_{n>=0} x^(n(n+1)/2) ], the log of the g.f. of A010054. - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 28 2008
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EXAMPLE
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a(28) = 40 because the sum of the even divisors of 28 (2, 4, 14 and 28) = 48 and the sum of the odd divisors of 28 (1 and 7) = 8; their absolute difference being 40.
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MATHEMATICA
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f[ n_Integer ] := (c = Divisors[ n ]; Abs[ Apply[ Plus, Select[ c, EvenQ[ # ] & ] ] - Apply[ Plus, Select[ c, OddQ[ # ] & ] ] ] ); Table[ f[ n ], {n, 1, 75} ]
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PROGRAM
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(PARI) a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*d))
(PARI) {a(n)=n*polcoeff(log(sum(k=0, (sqrtint(8*n+1)-1)\2, x^(k*(k+1)/2))+x*O(x^n)), n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 28 2008
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CROSSREFS
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A diagonal of A060044.
a(2^n)=-A036563(n+1). a(3^n)=A003462(n+1).
First differences of -A024919(n).
Cf. A010054.
Sequence in context: A096291 A016719 A090370 this_sequence A113184 A136004 A134299
Adjacent sequences: A002126 A002127 A002128 this_sequence A002130 A002131 A002132
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Better description and more terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 14 2000. More terms from N. J. A. Sloane (njas(AT)research.att.com), Mar 19, 2001.
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