|
| |
|
|
A002129
|
|
Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.
(Formerly M3236 N1307)
|
|
23
| |
|
|
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; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| 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).
|
|
|
REFERENCES
| 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).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
S. R. Finch, The "One-Ninth" Constant
|
|
|
FORMULA
| 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
Dirichlet g.f. zeta(s)*zeta(s-1)*(1-4/2^s). Dirichlet convolution of A000203 and the quasi-finite (1,-4,0,0,0,...). - R. J. Mathar, Mar 04 2011
a(n) = A000593(n)-A146076(n). - R. J. Mathar, Mar 05 2011
|
|
|
EXAMPLE
| 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.
|
|
|
MAPLE
| A002129 := proc(n) -add((-1)^d*d, d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 05 2011
|
|
|
MATHEMATICA
| f[n_] := Block[{c = Divisors@ n}, Plus @@ Select[c, EvenQ] - Plus @@ Select[c, OddQ]]; Array[f, 64] (* Robert G. Wilson v, Mar 04 2011 *)
|
|
|
PROG
| (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
|
|
|
CROSSREFS
| 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: A016719 A196999 A090370 * A113184 A136004 A134299
Adjacent sequences: A002126 A002127 A002128 * A002130 A002131 A002132
|
|
|
KEYWORD
| sign,easy,nice,mult
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| 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.
|
| |
|
|
