A060044 -id:A060044 - cp's OEIS frontend

A060045 Generalized sum of divisors function: third diagonal of A060044.

1, -1, 1, -2, 10, -11, 12, -21, 31, -13, 23, -42, 42, -42, 43, 22, -14, 33, -126, 185, -273, 406, -387, 637, -945, 1092, -1389, 1841, -2358, 2852, -3023, 3876, -4953, 5593, -6321, 7581, -9222, 10241, -11205, 14021, -16247, 17710, -19858, 23015, -26705, 28908, -31318, 36270, -41316, 45619, -49015, 55287

Offset: 9

N. J. A. Sloane, Mar 19 2001

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.

G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum(x^(n*i)/(1+x^n)^(2*i),n=1..inf), i=1..3. - Vladeta Jovovic, Sep 21 2007

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007

A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Glaisher calls this zeta(n) or zeta_1(n). - N. J. A. Sloane, Nov 24 2018
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).
Unsigned sequence is A113184. - Peter Bala, Dec 14 2020

			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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 3rd formula.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
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 = 1..10000 (terms 1..1000 from T. D. Noe)
Steven R. Finch, The "One-Ninth" Constant [Broken link]
Steven R. Finch, The "One-Ninth" Constant [From the Wayback machine]
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).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
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.
Index entries for sequences mentioned by Glaisher

A diagonal of A060044.
a(2^n) = -A036563(n+1). a(3^n) = A003462(n+1).
First differences of -A024919(n).
Cf. A010054, A113184.

A002129 := proc(n) -add((-1)^d*d,d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 05 2011

f[n_] := Block[{c = Divisors@ n}, Plus @@ Select[c, EvenQ] - Plus @@ Select[c, OddQ]]; Array[f, 64] (* Robert G. Wilson v, Mar 04 2011 *)
a[n_] := DivisorSum[n, -(-1)^#*#&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 3 - 2^(e + 1), (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]);  Array[a, 64] (* Amiram Eldar, Jul 20 2019 *)

PARI
```
a(n)=if(n<1,0,-sumdiv(n,d,(-1)^d*d))
```

{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, Jun 28 2008

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, 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, 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
a(n) = Sum_{j = 1..n} Sum_{k = 1..j} (-1)^(j+1)*cos(2*k*n*Pi/j). - Peter Bala, Aug 24 2022
G.f.: Sum_{n>=1} n*(-x)^(n-1)/(1-x^n). - Mamuka Jibladze, Jun 03 2025

Better description and more terms from Robert G. Wilson v, Dec 14 2000

More terms from N. J. A. Sloane, Mar 19 2001

A060177 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 5, 2, 1, 6, 4, 2, 11, 2, 5, 13, 4, 10, 17, 3, 1, 15, 22, 4, 2, 25, 27, 2, 5, 37, 29, 6, 10, 52, 37, 2, 20, 67, 44, 4, 1, 30, 97, 44, 4, 2, 52, 117, 55, 5, 5, 77, 154, 59, 2, 10, 117, 184, 68, 6, 20, 162, 235, 71, 2, 36, 227, 277, 81, 6, 1, 58, 309, 338

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  2  2  3  2  4  2  4  3  4  2  6 ...
        1  2  5  6 11 13 17 22 27 29 ...
                 1  2  5 10 15 25 37 ...
                             1  2  5 ...

Alois P. Heinz, Rows n = 1..500, flattened
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.

Diagonals give A000005, A002133, A002134, A365630, A365631. Cf. A060043, A060044.

Cf. A116608 (reflected rows). - Alois P. Heinz, Jan 29 2014

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1) +x*add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(coeff(p, x, degree(p)-k), k=0..degree(p)-1))(b(n$2)):
seq(T(n), n=1..25);  # Alois P. Heinz, Jan 29 2014

Reverse /@ Table[Length /@ Split[ Sort[Map[Length, Split /@ IntegerPartitions[n], {1}]]], {n, 24}] (* Wouter Meeussen, Apr 21 2012, updated by Jean-François Alcover, Jan 29 2014 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A060177_gen(): # generator of terms
    return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(isqrt((n<<3)+1)-1>>1,0,-1))
A060177_list = list(islice(A060177_gen(),30)) # Chai Wah Wu, Sep 15 2023

T(n,k) = Partitions of n using only k types of piles. Also, Sum_{k=1..A003056(n)} T(n,k)*k = A000070(n). Also, Sum_{k=1..A003056(n)} T(n,k)*(k-1) = A058884(n). - Naohiro Nomoto, Jan 24 2002

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 Naohiro Nomoto, Jan 24 2002

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

A060047 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 2, 4, 1, 4, 2, 6, 4, 8, 8, 8, 14, 8, 1, 18, 13, 2, 28, 12, 4, 40, 12, 8, 52, 16, 14, 70, 14, 24, 88, 16, 40, 104, 24, 1, 56, 140, 16, 2, 84, 168, 18, 4, 122, 196, 26, 8, 168, 240, 20, 14, 232, 278, 24, 24, 312, 320, 32, 40, 408, 380, 24, 64, 528, 440, 24, 100, 672, 504

Offset: 1

N. J. A. Sloane, Mar 19 2001

Lengths of rows are 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 ... (A000196).

			Triangle turned on its side begins:
  1  2  4  4  6  8  8  8 13 12 12 ...
           1  2  4  8 14 18 28 40 ...
                          1  2  4 ...
For example, T(6,1) = 8, T(6,2) = 4.

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.

Diagonals give A002131, A002132, A060046, A365666, A365667.

Cf. A060043, A060044, A060177, A060184.

T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*(2*m_1-1) + s_2*(2*m_2-1) + ... + s_k*(2*m_k-1) = 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^(2*m_1+2*m_2+...+2*m_k-k)/((1-q^{2*m_1-1})*(1-q^{2*m_2-1})*...*(1-q^{2*m_k-1}))^2 = Sum_n T(n, k)*q^n.
G.f. for k-th diagonal: (-1)^k * (1/k) * ( Sum_{j>=k} (-1)^j * j * binomial(j+k-1,2*k-1) * q^(j^2) ) / ( 1 + 2 * Sum_{j>=1} (-q)^(j^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto, Jan 24 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

A002130 Generalized sum of divisors function.

Original entry on oeis.org

1, -1, 1, 3, -2, 1, -5, 23, -25, 27, -49, 74, -62, 85, -132, 165, -195, 229, -240, 325, -374, 379, -469, 553, -590, 746, -805, 854, -1000, 1085, -1168, 1284, -1396, 1668, -1767, 1815, -2030, 2297, -2450, 2480, -2849, 3293, -3113, 3278, -3772, 4091, -4230, 4213, -4830, 5607, -5499, 5430, -6018, 6922, -6880

Offset: 3

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).

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 A060044.

terms = 55; offset = 3; t[i_] := Sum[x^(n*i)/(1 + x^n)^(2*i), {n, 1, terms + 5}]; s = Series[(t[1]^2 - t[2])/2, {x, 0, terms + 5 }]; A002130 = CoefficientList[s, x][[offset + 1 ;; terms + offset]] (* Jean-François Alcover, Dec 11 2014, after Vladeta Jovovic *)

G.f.: (t(1)^2-t(2))/2 where t(i) = Sum_{n>=1} x^(n*i)/(1+x^n)^(2*i), i=1..2. - Vladeta Jovovic, Sep 21 2007

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007

cp's OEIS Frontend

A060045 Generalized sum of divisors function: third diagonal of A060044.

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A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.

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A060177 Triangle of generalized sum of divisors function, read by rows.

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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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A060047 Triangle of generalized sum of divisors function, read by rows.

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

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A002130 Generalized sum of divisors function.

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