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 A002131 Sum of divisors d of n such that n/d is odd. (Formerly M0937 N0351) 41

%I M0937 N0351

%S 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,

%T 28,40,32,30,48,32,32,48,36,48,52,38,40,56,48,42,64,44,48,78,48,48,64,

%U 57,62,72,56,54,80,72,64,80,60,60,96,62,64,104,64,84,96,68,72,96,96,72

%N Sum of divisors d of n such that n/d is odd.

%C Glaisher calls this Delta'(n) or Delta'_1(n). - _N. J. A. Sloane_, Nov 24 2018

%C Equals row sums of triangle A143119. - _Gary W. Adamson_, Jul 26 2008

%C 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

%C 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

%C From _Omar E. Pol_, Nov 26 2019: (Start)

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

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

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

%D A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 294, Art. 386.

%D 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)

%D 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).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002131/b002131.txt">Table of n, a(n) for n = 1..10000</a>

%H H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H G.-H. Halphen, <a href="http://www.numdam.org/item?id=BSMF_1878__6__119_1">Sur les sommes des diviseurs des nombres entiers et les décompositions en deux carrés</a>, Bull. math. Soc. France, 6 (1877-1878), 119-120.

%H P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

%F 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

%F 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

%F 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

%F 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

%F 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

%F G.f.: Sum_{n>=1} n*x^n/(1-x^(2*n)). - _Vladeta Jovovic_, Oct 16 2002

%F G.f.: Sum_{k>0} x^(2*k - 1) / (1 - x^(2*k - 1))^2. - _Michael Somos_, Aug 17 2005

%F 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.

%F 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

%F Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 1/2^s). - _Michael Somos_, Apr 05 2003

%F Moebius transform is A026741.

%F 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

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

%F a(n) = A006519(n)*A000203(n/A006519(n)). - _Robert Israel_, Jul 05 2016

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 /16. - _Vaclav Kotesovec_, Feb 01 2019

%F a(n) = (A000203(n) + A000593(n))/2. - _Amiram Eldar_, Aug 12 2019

%F From _Peter Bala_, Jan 06 2021: (Start)

%F G.f.: A(x) = (1/2)*Sum_{n = -oo..oo} x^(2*n+1)/(1 - x^(2*n+1))^2.

%F A(x) = Sum_{n = -oo..oo} x^(4*n+1)/(1 - x^(4*n+1))^2.

%F a(2*n) = 2*a(n); a(2*n+1) = A008438(n). (End)

%e 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 + ...

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

%e 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

%e 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

%p a:= proc(n) local e;

%p e*numtheory:-sigma(n/e)

%p end proc:

%p map(a, [\$1..100]); # _Robert Israel_, Jul 05 2016

%t a[n_]:=Total[Cases[Divisors[n], d_ /; OddQ[n/d]]]; Table[a[n],{n,1,71}] (* _Jean-François Alcover_, Mar 18 2011 *)

%t a[ n_] := If[ n < 1, 0, DivisorSum[n, # / GCD[#, 2] &]] (* _Michael Somos_, Aug 01 2011 *)

%t 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 *)

%t Table[Total[Select[Divisors[n],OddQ[n/#]&]],{n,80}] (* _Harvey P. Dale_, Jun 05 2015 *)

%t 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 *)

%t f[2, e_] := 2^e; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2020 *)

%o (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 */

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d / gcd(d, 2)))}; /* _Michael Somos_, Apr 05 2003 */

%o (PARI) a(n) = my(v = valuation(n, 2)); sigma(n>>v)<<v \\ _David A. Corneth_, Aug 12 2019

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

%o -- _Reinhard Zumkeller_, Aug 14 2011

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

%Y A diagonal of A060047. Bisection A008438.

%Y Cf. A000203, A000593, A006519, A026741, A143119, A192065, A244051, A301798, A326938 (Dirichlet inverse).

%K nonn,nice,easy,mult

%O 1,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified April 18 05:11 EDT 2021. Contains 343072 sequences. (Running on oeis4.)