A002128 -id:A002128 - cp's OEIS frontend

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

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

A002127 MacMahon's generalized sum of divisors function.

Original entry on oeis.org

1, 3, 9, 15, 30, 45, 67, 99, 135, 175, 231, 306, 354, 465, 540, 681, 765, 945, 1040, 1305, 1386, 1695, 1779, 2205, 2290, 2754, 2835, 3438, 3480, 4185, 4272, 5076, 5004, 6100, 5985, 7155, 7154, 8325, 8190, 9840, 9471, 11241, 11055, 12870, 12420, 14911

Offset: 3

N. J. A. Sloane

Number of partitions of n with two designated summands. For example: a(5) = 9 because there are 9 partitions of 5 with two designated summands: [4'+ 1'], [3'+ 2'], [3'+ 1'+ 1], [3'+ 1 + 1'], [2'+ 2 + 1'], [2 + 2'+ 1'], [2'+ 1'+ 1 + 1], [2'+ 1 + 1'+ 1], [2'+ 1 + 1 + 1']. - Omar E. Pol, Jul 23 2025

			x^3 + 3*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 45*x^8 + 67*x^9 + 99*x^10 + ...

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).

John Cerkan, Table of n, a(n) for n = 3..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.
William Craig, Jan-Willem van Ittersum and Ken Ono, Integer partitions detect the primes, PNAS, Vol. 121, No. 39 (2024), e2409417121.
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.
S. Rose, What literature is known about MacMahon's generalized sum-of-divisors function?

A diagonal of A060043.
Cf. A002128.
Column 2 of A385001.

A002127[n_] := (DivisorSigma[3, n] - (2*n - 1)*DivisorSigma[1, n])/8;
Array[A002127, 50, 3] (* Paolo Xausa, Jul 04 2025, after Michael Somos's PARI *)

{a(n) = if( n<1, 0, ( sigma( n, 3) - (2*n - 1) * sigma(n) ) / 8)} /* Michael Somos, Jan 10 2012 */

G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+2, 4) * x^( k*(k+1) / 2 )) / (5  * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - Michael Somos, Jan 10 2012
a(n) = (n^2 - 3*n + 2) * A000203(n) / 8 iff n is an odd prime (see Craig link et al.).
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (4!*5!). - Vaclav Kotesovec, Aug 01 2025

More terms from Vladeta Jovovic, Nov 11 2001

A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 4, 1, 0, 7, 3, 0, 6, 9, 0, 12, 15, 1, 0, 8, 30, 3, 0, 15, 45, 9, 0, 13, 67, 22, 0, 18, 99, 42, 1, 0, 12, 135, 81, 3, 0, 28, 175, 140, 9, 0, 14, 231, 231, 22, 0, 24, 306, 351, 51, 0, 24, 354, 551, 97, 1, 0, 31, 465, 783, 188, 3, 0, 18, 540, 1134, 330, 9

Offset: 0

Omar E. Pol, Jul 17 2025

The divisor function sigma_1(n) = A000203(n) is also the number of partitions of n with only one designated summand, n >= 1.
When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Alternating row sums give A329157.
Columns converge to A000716.
This triangle equals A060043 with reversed rows and an additional column 0.

			Triangle begins:
--------------------------------------------
   n\k:   0    1     2     3     4    5   6
--------------------------------------------
   0 |    1;
   1 |    0,   1;
   2 |    0,   3;
   3 |    0,   4,    1;
   4 |    0,   7,    3;
   5 |    0,   6,    9;
   6 |    0,  12,   15,    1;
   7 |    0,   8,   30,    3;
   8 |    0,  15,   45,    9;
   9 |    0,  13,   67,   22;
  10 |    0,  18,   99,   42,    1;
  11 |    0,  12,  135,   81,    3;
  12 |    0,  28,  175,  140,    9;
  13 |    0,  14,  231,  231,   22;
  14 |    0,  24,  306,  351,   51;
  15 |    0,  24,  354,  551,   97,   1;
  16 |    0,  31,  465,  783,  188,   3;
  17 |    0,  18,  540, 1134,  330,   9;
  18 |    0,  39,  681, 1546,  568,  22;
  19 |    0,  20,  765, 2142,  918,  51;
  20 |    0,  42,  945, 2835, 1452, 108;
  21 |    0,  32, 1040, 3758, 2233, 208,  1;
  ...
For n = 6 and k = 1 there are 12 partitions of 6 with only one designated summand as shown below:
   6'
   3'+ 3
   3 + 3'
   2'+ 2 + 2
   2 + 2'+ 2
   2 + 2 + 2'
   1'+ 1 + 1 + 1 + 1 + 1
   1 + 1'+ 1 + 1 + 1 + 1
   1 + 1 + 1'+ 1 + 1 + 1
   1 + 1 + 1 + 1'+ 1 + 1
   1 + 1 + 1 + 1 + 1'+ 1
   1 + 1 + 1 + 1 + 1 + 1'
So T(6,1) = 12, the same as A000203(6) = 12.
.
For n = 6 and k = 2 there are 15 partitions of 6 with two designated summands as shown below:
   5'+ 1'
   4'+ 2'
   4'+ 1'+ 1
   4'+ 1 + 1'
   3'+ 1'+ 1 + 1
   3'+ 1 + 1'+ 1
   3'+ 1 + 1 + 1'
   2'+ 2 + 1'+ 1
   2'+ 2 + 1 + 1'
   2 + 2'+ 1'+ 1
   2 + 2'+ 1 + 1'
   2'+ 1'+ 1 + 1 + 1
   2'+ 1 + 1'+ 1 + 1
   2'+ 1 + 1 + 1'+ 1
   2'+ 1 + 1 + 1 + 1'
So T(6,2) = 15, the same as A002127(6) = 15.
.
For n = 6 and k = 3 there is only one partition of 6 with three designated summands as shown below:
   3'+ 2'+ 1'
So T(6,3) = 1, the same as A002128(6) = 1.
There are 28 partitions of 6 with designated summands, so A077285(6) = 28.
.

Alois P. Heinz, Rows n = 0..1000, flattened

Columns: A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5), A384926 (k=6).
Row sums give A077285.
Cf. A000217, A002024, A003056, A060043, A196020, A252117, A329157, A365434, A384999.
Cf. A000096, A000716, A116608, A293421, A384998, A384999.

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,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 18 2025

From Alois P. Heinz, Jul 18 2025: (Start)
Sum_{k>=1} k * T(n,k) = A293421(n).
T(A000096(n),n) = A000716(n). (End)
G.f.: Product_{i>0} 1 + (y*x^i)/(1 - x^i)^2. - John Tyler Rascoe, Jul 23 2025
Conjecture: For fixed k >= 1, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / ((2*k)! * (2*k+1)!). - Vaclav Kotesovec, Aug 01 2025

A365664 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 97, 188, 330, 568, 918, 1452, 2233, 3344, 4884, 7004, 9856, 13653, 18699, 25080, 33462, 43918, 57304, 73668, 94482, 119262, 150285, 187231, 232560, 285660, 350746, 425627, 516477, 620731, 745503, 887796, 1056669, 1247521, 1472460, 1726054, 2021327

Offset: 10

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with four designated summands. For example: a(11) = 3 because there are three partitions of 11 with four designated summands: [5'+ 3'+ 2'+ 1'], [4'+ 3'+ 2'+ 1'+ 1], [4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 26 2025

Seiichi Manyama, Table of n, a(n) for n = 10..10000
George E. Andrews and Simon C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2013, No. 676 (2013), pp. 97-103; arXiv preprint, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365665.
Cf. A010816, A365630, A365666.
Cf. A001158, A001160, A013955.
Column k=4 of A385001.

a[n_] := Module[{d = DivisorSigma[{1, 3, 5, 7}, n]}, (5*d[[4]] - (126*n-441)*d[[3]] + (756*n^2-4410*n+4935)*d[[2]] - (840*n^3-5880*n^2+9870*n-3229)*d[[1]])/967680]; Array[a, 40, 10] (* Amiram Eldar, Jan 07 2025 *)

a(n) = (5*sigma(n, 7)-(126*n-441)*sigma(n, 5)+(756*n^2-4410*n+4935)*sigma(n, 3)-(840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680; \\ Seiichi Manyama, Jul 24 2024

G.f.: (1/9) * ( Sum_{k>=4} (-1)^k * (2*k+1) * binomial(k+4,8) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
a(n) = (5*sigma_7(n) - (126*n-441)*sigma_5(n) + (756*n^2-4410*n+4935)*sigma_3(n) - (840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680. - Seiichi Manyama, Jul 24 2024
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / (8!*9!). - Vaclav Kotesovec, Aug 01 2025

A365665 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 208, 390, 693, 1193, 1977, 3195, 4995, 7722, 11583, 17164, 24882, 35685, 50205, 70083, 96300, 131101, 176358, 235377, 310651, 407352, 529074, 682750, 874038, 1112085, 1405521, 1766259, 2206413, 2741431, 3389052, 4168089, 5103450, 6218469

Offset: 15

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with five designated summands (when part i has multiplicity j > 0 exactly one part i is "designated"). For example: a(16) = 3 because there are three partitions of 16 with five designated summands: [6'+ 4'+ 3'+ 2'+ 1'], [5'+ 4'+ 3'+ 2'+ 1'+ 1], [5'+ 4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 29 2025

Vaclav Kotesovec, Table of n, a(n) for n = 15..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.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365664.
Cf. A010816, A365631, A365667.
Column k=5 of A385001.
Cf. A384926.

nmax = 60; Drop[CoefficientList[Series[-1/11 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 5, 10]*x^(k*(k + 1)/2), {k, 5, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x], 15] (* Vaclav Kotesovec, Jul 29 2025 *)
(* or *)
Table[(10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080) * DivisorSigma[1, n] + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080) * DivisorSigma[3, n] + (133/1228800 - n/20480 + n^2/215040) * DivisorSigma[5, n] + (1/516096 - n/3096576) * DivisorSigma[7, n] + DivisorSigma[9, n]/154828800, {n, 15, 60}] (* Vaclav Kotesovec, Jul 29 2025 *)

G.f.: -(1/11) * ( Sum_{k>=5} (-1)^k * (2*k+1) * binomial(k+5,10) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
From Vaclav Kotesovec, Jul 29 2025: (Start)
a(n) = (10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080)*sigma(n) + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080)*sigma_3(n) + (133/1228800 - n/20480 + n^2/215040)*sigma_5(n) + (1/516096 - n/3096576)*sigma_7(n) + sigma_9(n)/154828800.
Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 144850083840000.
(End)

A374930 Expansion of Sum_{1<=i<=j<=k} q^(i+j+k)/( (1-q^i)(1-q^j)(1-q^k) )^2.

Original entry on oeis.org

1, 7, 27, 77, 181, 378, 707, 1254, 2052, 3290, 4928, 7371, 10381, 14756, 19818, 27158, 35139, 46683, 58806, 76146, 93555, 119092, 143222, 178983, 212408, 261261, 305046, 371931, 428156, 515592, 589385, 701442, 792720, 939918, 1050567, 1233387, 1374835, 1600143, 1766583, 2052898, 2247784

Offset: 3

Seiichi Manyama, Jul 24 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 3..10000
Tewodros Amdeberhan, George E. Andrews and Roberto Tauraso, Extensions of MacMahon's sums of divisors, Research in the Mathematical Sciences, Vol. 11 (2024), Article number 8; arXiv preprint, arXiv:2309.03191 [math.CO], 2023.

Cf. A374929, A374931.

Cf. A000203, A001158, A001160, A002128.

A374930[n_] := (31*DivisorSigma[5, n] - 70*(n + 1)*DivisorSigma[3, n] + (40*n^2 + 60*n + 9)*DivisorSigma[1, n])/1920;
Array[A374930, 50, 3] (* Paolo Xausa, Jul 24 2024 *)

a(n) = (31*sigma(n, 5)-70*(n+1)*sigma(n, 3)+(40*n^2+60*n+9)*sigma(n))/1920;

from math import prod
from sympy import factorint
def A374930(n):
    f = factorint(n).items()
    return (31*prod((p**(5*(e+1))-1)//(p**5-1) for p,e in f)-70*(n+1)*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f) + (20*n*((n<<1)+3)+9)*prod((p**(e+1)-1)//(p-1) for p, e in f))//1920 # Chai Wah Wu, Jul 24 2024

a(n) = (31*sigma_5(n) - 70*(n+1)*sigma_3(n) + (40*n^2+60*n+9)*sigma(n))/1920.

A384926 Number of partitions of n with six designated summands.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 414, 765, 1344, 2310, 3834, 6248, 9894, 15408, 23550, 35394, 52353, 76402, 109959, 156366, 219850, 305796, 421281, 574568, 777234, 1042083, 1387037, 1831362, 2402595, 3128995, 4051797, 5211639, 6668490, 8482089, 10737063, 13516615

Offset: 21

Omar E. Pol, Jul 23 2025

nonn

			21 has only one partition with six designated summands: [6'+ 5'+ 4'+ 3'+ 2'+ 1'], so a(21) = 1.
22 has three partitions with six designated summands: [7'+ 5'+ 4'+ 3'+ 2'+ 1'], [6'+ 5'+ 4'+ 3'+ 2'+ 1'+ 1], [6'+ 5'+ 4'+ 3'+ 2'+ 1 + 1'], so a(22) = 3.

Alois P. Heinz, Table of n, a(n) for n = 21..4000

Column k=6 of A385001.
Columns of A385001 converge to A000716.
Other columns k of A385001 are A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5).

b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j*x, j=1..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=21..57);  # Alois P. Heinz, Jul 23 2025

nmax=60; Drop[CoefficientList[Series[1/13 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 6, 12]*x^(k*(k + 1)/2), {k, 6, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x] , 21] (* Vaclav Kotesovec, Jul 29 2025 *)

A000716(n) >= a(21+n) with equality only for n <= 6.

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / (12! * 13!). - Vaclav Kotesovec, Aug 01 2025

More terms from Alois P. Heinz, Jul 23 2025

cp's OEIS Frontend

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

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A002127 MacMahon's generalized sum of divisors function.

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A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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A365664 Expansion of Sum_{0

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A365665 Expansion of Sum_{0

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A374930 Expansion of Sum_{1<=i<=j<=k} q^(i+j+k)/( (1-q^i)*(1-q^j)*(1-q^k) )^2.

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A384926 Number of partitions of n with six designated summands.

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A374930 Expansion of Sum_{1<=i<=j<=k} q^(i+j+k)/( (1-q^i)(1-q^j)(1-q^k) )^2.