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A002131 Sum of divisors d of n such that n/d is odd.
(Formerly M0937 N0351)
38
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; text; internal format)
OFFSET

1,2

COMMENTS

Glaisher calls this Delta'(n) or Delta'_1(n). - N. J. A. Sloane, Nov 24 2018

Equals row sums of triangle A143119. - Gary W. Adamson, 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

a(n) = 2*(a(n-1) - a(n-4) + a(n-9) ... +- a(n-i^2) ...) up to the last positive number n - i^2, and if n is a square, then a(0) should be replaced by n/2 (cf. Halphen). - Michel Marcus, Oct 14 2012

From Omar E. Pol, Nov 26 2019: (Start)

a(n) is also the total number of odd parts in the partitions of n into equal parts.

a(n) = n iff n is a power of 2.

a(n) = n + 1 iff n is an odd prime. (End)

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)

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).

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, arXiv:math/0407061 [math.NT], 2004.

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

G.-H. Halphen, Sur les sommes des diviseurs des nombres entiers et les décompositions en deux carrés, Bull. math. Soc. France, 6 (1877-1878), 119-120.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.

Index entries for sequences mentioned by Glaisher

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 complete 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, 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, 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, Oct 16 2002

G.f.: Sum_{k>0} x^(2*k - 1) / (1 - x^(2*k - 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 in Z} (-1)^k * q^(k^2)) where theta_4(u) is one of Jacobi's theta functions.

G.f.: A(q) = 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.

a(n) = n * Sum_{c|n} 1/c, where c are odd numbers (A005408) dividing n. a(n) = A069359(n) + n. a(n) = A000035(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d) = Sum_{d|n} A000035(d) * A000027(n/d). -Jaroslav Krizek, Nov 07 2013

L.g.f.: Sum_{ k>0 } atanh(x^k) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016

a(n) = A006519(n)*A000203(n/A006519(n)). - Robert Israel, Jul 05 2016

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 /16. - Vaclav Kotesovec, Feb 01 2019

a(n) = (A000203(n) + A000593(n))/2. - Amiram Eldar, Aug 12 2019

EXAMPLE

G.f. = q + 2*q^2 + 4*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 8*q^7 + 8*q^8 + 13*q^9 + ...

The divisors of 6 are 1, 2, 3, and 6. Only 6/2 and 6/6 are odd. Hence, a(6) = 2 + 6 = 8.

As 120 = 15 * 2^3 where 15 is odd and 2^3 is the largest power of 2 dividing 120, a(120) = sigma(15) * 2^3 = 24 * 8 = 192. - David A. Corneth, Aug 12 2019

For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1]. There are 8 odd parts, so a(6) = 8. - Omar E. Pol, Nov 26 2019

MAPLE

a:= proc(n) local e;

  e:= 2^padic:-ordp(n, 2);

  e*numtheory:-sigma(n/e)

end proc:

map(a, [$1..100]); # Robert Israel, Jul 05 2016

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 *)

Table[Total[Select[Divisors[n], OddQ[n/#]&]], {n, 80}] (* Harvey P. Dale, Jun 05 2015 *)

a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1/2) (EllipticK[ m] / Pi)^2 (D[ JacobiZeta[ JacobiAmplitude[x, m], m], x] /. x -> 0)], {q, 0, n}]; (* Michael Somos, Mar 17 2017 *)

PROG

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - (p<3) * X) / ((1 - X) * (1 - p*X))) [n])}; /* Michael Somos, Apr 05 2003 */

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d / gcd(d, 2)))}; /* Michael Somos, Apr 05 2003 */

(PARI) a(n) = my(v = valuation(n, 2)); sigma(n>>v)<<v \\ David A. Corneth, Aug 12 2019

(Haskell)

a002131 n = sum [d | d <- [1..n], mod n d == 0, odd $ div n d]

-- Reinhard Zumkeller, Aug 14 2011

(MAGMA) [&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..75]]; // Marius A. Burtea, Aug 12 2019

CROSSREFS

A diagonal of A060047.

Cf. A000203, A000593, A006519, A026741, A143119, A192065, A244051, A301798.

Sequence in context: A053196 A159634 A186690 * A230641 A063200 A063224

Adjacent sequences:  A002128 A002129 A002130 * A002132 A002133 A002134

KEYWORD

nonn,nice,easy,mult

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)