A060047 -id:A060047 - cp's OEIS frontend

A002131 Sum of divisors d of n such that n/d is odd.

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

Offset: 1

N. J. A. Sloane, Simon Plouffe

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)

			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

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).

T. D. Noe, Table of n, a(n) for n = 1..10000
Heng Huat Chan and Christian 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.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025. See p. 2.
Index entries for sequences mentioned by Glaisher

A diagonal of A060047. Bisection A008438.

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

a002131 n = sum [d | d <- [1..n], mod n d == 0, odd $ div n d]
-- Reinhard Zumkeller, Aug 14 2011

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

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

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 *)
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 *)

{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 */

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

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

from math import prod
from sympy import factorint
def A002131(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Dec 17 2021

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
From Peter Bala, Jan 06 2021: (Start)
G.f.: A(x) = (1/2)*Sum_{n = -oo..oo} x^(2*n+1)/(1 - x^(2*n+1))^2.
A(x) = Sum_{n = -oo..oo} x^(4*n+1)/(1 - x^(4*n+1))^2.
a(2*n) = 2*a(n); a(2*n+1) = A008438(n). (End)
Expansion of (-1/2) x (d phi(-x) / dx) / phi(-x) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jul 01 2023

A060043 Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 3, 1, 4, 3, 7, 9, 6, 1, 15, 12, 3, 30, 8, 9, 45, 15, 22, 67, 13, 1, 42, 99, 18, 3, 81, 135, 12, 9, 140, 175, 28, 22, 231, 231, 14, 51, 351, 306, 24, 1, 97, 551, 354, 24, 3, 188, 783, 465, 31, 9, 330, 1134, 540, 18, 22, 568, 1546, 681, 39, 51, 918, 2142, 765, 20

Offset: 1

N. J. A. Sloane, Mar 19 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).

			Triangle turned on its side begins:
1 3 4 7 6 12  8 15 13 18 ...
    1 3 9 15 30 45 67 99 ...
           1  3  9 22 42 ...
                       1 ...
For example, T(6,2) = 15.

Alois P. Heinz, Rows n = 1..1000, flattened
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A000203, A002127, A002128, A365664, A365665.

Cf. A010816, A060044, A060047.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, degree(p)-i), i=0..degree(p)-1))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Jul 21 2025

Clear[diag, m]; nmax = 19; kmax = Floor[(Sqrt[8*nmax+1]-1)/2]; m[0] = 0; diag[k_] := diag[k] = Sum[q^(Sum[m[i], {i, 1, k}])/(Times @@ (1 - q^Array[m, k]))^2, Sequence @@ Table[{m[j], m[j-1]+1, nmax}, {j, 1, k}] // Evaluate] + O[q]^(nmax+1) // CoefficientList[#, q]&; Table[ Select[ Table[diag[k][[j+1]], {k, 1, kmax}], IntegerQ[#] && # > 0&] // Reverse, {j, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

T(n, 1) = sum of divisors of n (A000203), T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.
G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k))^2 = Sum_n T(n, k)*q^n.
G.f. for k-th diagonal: (-1)^k * (1/(2*k+1)) * ( Sum_{j>=k} (-1)^j * (2*j+1) * binomial(j+k,2*k) * q^(j*(j+1)/2) ) / ( Sum_{j>=0} (-1)^j * (2*j+1) * q^(j*(j+1)/2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto, Jan 24 2002

A002132 Generalized sum of divisors function.

Original entry on oeis.org

1, 2, 4, 8, 14, 18, 28, 40, 52, 70, 88, 104, 140, 168, 196, 240, 278, 320, 380, 440, 504, 562, 644, 720, 808, 910, 1000, 1120, 1240, 1360, 1488, 1600, 1789, 1938, 2100, 2296, 2452, 2660, 2880, 3080, 3292, 3542, 3784, 4048, 4400, 4572, 4868, 5280, 5502, 5850

Offset: 4

N. J. A. Sloane

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).

Seiichi Manyama, Table of n, a(n) for n = 4..10000
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

A diagonal of A060047.

Cf. A015128.

nmax = 60; Drop[CoefficientList[Series[1/2 * Sum[(-1)^k*k*Binomial[k + 1, 3]*x^(k^2), {k, 2, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 4] (* Vaclav Kotesovec, Jul 30 2025 *)

G.f.: (1/2) * ( Sum_{k>=2} (-1)^k * k * binomial(k+1,3) * q^(k^2) ) / ( 1 + 2 * Sum_{k>=1} (-q)^(k^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto and Vladeta Jovovic, Jan 25 2002

A060184 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 0, -1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 1, -2, 0, 1, 3, 1, 5, 6, 0, 0, -1, -1, 2, 1, 5, 5, -2, 0, -2, -3, 2, 2, 9, 10, 0, 1, 4, 3, 0, 4, 0, 2, 9, 9, -3, 1, 3, -2, -7, 2, 0, 3, 14, 16, 0, 2, 6, -1, -9, 2, 0, 3, 15, 17, -2, 1, 8, 19, 10, -6, 4, 0, -1, 0, 15, 22, 0, 1, 9, 21, 7, -13, 2, 0, -2, -4, 11, 20, -4, 2, 15, 33, 14, -15, 3, 0, -4, -10, 10, 28, 0, 3

Offset: 1

N. J. A. Sloane, Mar 20 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
  1  0  2 -1  2  0  2 -2  3  0  2 ...
        1  0  1  2  1  1  1  6 -1 ...
              1  0  1  0  5 -1  5 ...

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A048272, A060185, A060186.

Cf. A060043, A060044, A060047, A060177.

max = 27(*rows*); t[n_, k_] := Module[{m, mm, q, s}, mm = Array[m, k]; s = Sum[q^Total[mm]/Times @@ (1+q^mm), Evaluate[Sequence @@ Transpose[{mm, Join[{1}, Most[mm]+1], max-Range[k-1, 0, -1]}]]]; SeriesCoefficient[s, {q, 0, n}]]; Table[Print[an = Table[t[n, k], {k, Floor[(Sqrt[8*n+1]-1)/2], 1, -1}]]; an, {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 21 2014 *)

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1+q^m_1)*(1+q^m_2)*...*(1+q^m_k)) = Sum_n T(n, k)*q^n.

More terms from Vladeta Jovovic, Sep 20 2007

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A060046 Generalized sum of divisors function: third diagonal of A060047.

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A002131 Sum of divisors d of n such that n/d is odd.

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A060043 Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.

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A002132 Generalized sum of divisors function.

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A060184 Triangle of generalized sum of divisors function, read by rows.

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A365666 Expansion of Sum_{0

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A365667 Expansion of Sum_{0

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