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A002131
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Sum of divisors d of n such that n/d is odd.
(Formerly M0937 N0351)
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7
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1, 2, 4, 4, 6, 8, 8, 8, 13, 12, 12, 16, 14, 16, 24, 16, 18, 26, 20, 24, 32, 24, 24, 32, 31, 28, 40, 32, 30, 48, 32, 32, 48, 36, 48, 52, 38, 40, 56, 48, 42, 64, 44, 48, 78, 48, 48, 64, 57, 62, 72, 56, 54, 80, 72, 64, 80, 60, 60, 96, 62, 64, 104, 64, 84, 96, 68, 72, 96, 96, 72
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Equals row sums of triangle A143119. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 26 2008
Cayley begins article 386 with "To find the value of A, = 8{q/(1-q)^2 + q^3/(1-q^3)^2 +&c.}," where A is 8 time the g.f. of this sequence. - Michael Somos, Aug 01 2011
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REFERENCES
| A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 294, Art. 386.
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066)
G.-H. Halphen, Sur les sommes des diviseurs des nombres entiers et les decompositions en deus carres, Bull. math. Soc. France, 6 (1877-1878), 119-.
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.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eqs. (5.1.29.3),(5.1.29.9)
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
| T. D. Noe, Table of n, a(n) for n=1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares
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FORMULA
| Expansion of K(k^2) * (K(k^2) - E(k^2)) / (2 * pi^2) in powers of q where q is Jacobi's nome and K(), E() are elliptic integrals. - Michael Somos, Aug 01 2011
Multiplicative with a(p^e) = p^e if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
a(n)=sigma(n)-sigma(n/2) for even n and =sigma(n) otherwise where sigma(n) is the sum of divisors of n (A000203). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Apr 07 2002
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u1*u6 - u1*u3 - 10*u2*u6 + u2^2 + 2*u2*u3 + 9*u6^2 - Michael Somos, Apr 10 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u2 - 3*u6)^2 - (u1 - 2*u2) * (u3 - 2*u6) - Michael Somos, Sep 06 2005
G.f.: Sum_{n>=1} n*x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 16 2002
G.f.: Sum_{k>0} x^(2k - 1) / (1 - x^(2k - 1))^2. - Michael Somos, Aug 17 2005
G.f.: (1/8) * theta_4''(0) / theta_4(0) = (Sum_{k>0} -(-1)^k * k^2 q^(k^2)) / (Sum_{k} (-1)^k * q^(k^2)) where theta_4(u) is one of Jacobi's theta functions.
G.f.: Z'(0) K^2 / (2 Pi^2) = (K - E) * K /(2 Pi^2) where Z(u) is the Jacobi Zeta function and K, E are complete elliptic integrals. - Michael Somos, Sep 06 2005
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/2^s). - Michael Somos, Apr 05 2003
Moebius transform is A026741.
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MATHEMATICA
| a[n_]:=Total[Cases[Divisors[n], d_ /; OddQ[n/d]]]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Mar 18 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, # / GCD[#, 2] &]] (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) EllipticK[ m] ( EllipticK[ m] - EllipticE[ m] ) / (Pi/2 )^2, {q, 0, n}]] (* Michael Somos, Aug 01 2011 *)
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PROG
| (PARI) {a(n) = direuler( p=2, n, (1 - (p<3) * X) / (1 - X) / (1 - p*X))[n]}
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d / gcd(d, 2)))}
(Haskell)
a002131 n = sum [d | d <- [1..n], mod n d == 0, odd $ div n d]
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
| A diagonal of A060047.
Cf. A000203, A026741, A143119.
Sequence in context: A053196 A159634 A186690 * A063200 A063224 A023847
Adjacent sequences: A002128 A002129 A002130 * A002132 A002133 A002134
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KEYWORD
| nonn,nice,easy,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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