A007437 -id:A007437 - cp's OEIS frontend

A366135 Expansion of Sum_{k>=1} k^3 * x^k/(1 - x^k)^3.

1, 11, 33, 98, 140, 366, 371, 820, 936, 1550, 1397, 3276, 2288, 4102, 4650, 6696, 5066, 10413, 7049, 13860, 12306, 15422, 12443, 27480, 17825, 25246, 25650, 36652, 24824, 51900, 30287, 54096, 46266, 55862, 52150, 93366, 51356, 77710, 75738, 116200, 69782, 137172

Offset: 1

Seiichi Manyama, Oct 28 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007437, A309731, A309732.
Cf. A001158, A328259.
Cf. A000203, A001157.

a[n_] := (n * DivisorSigma[1, n] + DivisorSigma[2, n]) * n/2; Array[a, 50] (* Amiram Eldar, Dec 15 2023 *)

PARI
```
a(n) = n*(n*sigma(n)+sigma(n, 2))/2;
```

a(n) = n * (n * sigma(n) + sigma_2(n))/2.
a(n) = Sum_{d|n} d^3 * binomial(n/d+1,2).
a(n) = Sum_{k=1..n} k*sigma_2(gcd(n,k)).
Sum_{k=1..n} a(k) ~ (Pi^2/48 + zeta(3)/8) * n^4. - Amiram Eldar, Dec 15 2023

A278947 Expansion of Sum_{k>=1} (k(5k - 3)/2)*x^k/(1 - x^k).

Original entry on oeis.org

0, 1, 8, 19, 42, 56, 107, 113, 190, 208, 298, 287, 483, 404, 589, 614, 806, 698, 1079, 875, 1302, 1202, 1471, 1289, 2035, 1581, 2062, 1990, 2541, 2060, 3142, 2357, 3318, 2978, 3544, 3178, 4641, 3368, 4435, 4166, 5390, 4142, 6106, 4559, 6279, 5798, 6517, 5453, 8339, 6042, 7998, 7142, 8778, 6944, 10070, 7822, 10445, 8930, 10390

Offset: 0

Ilya Gutkovskiy, Dec 02 2016

Inverse Moebius transform of heptagonal numbers (A000566).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Heptagonal Number.
Index to sequences related to polygonal numbers.

Cf. A000203, A000566 (heptagonal numbers), A002117, A059358.

Inverse Moebius transforms of polygonal numbers: A007437 (k=3), A001157 (k=4), A116913 (k=5), A278945 (k=6), this sequence (k=7).

nmax=58; CoefficientList[Series[Sum[(k (5 k - 3)/2) x^k/(1 - x^k),  {k, 1, nmax}], {x, 0, nmax}], x]
Flatten[{0, Table[(5*DivisorSigma[2, n] - 3*DivisorSigma[1, n])/2, {n, 1, 100}]}] (* Vaclav Kotesovec, Dec 05 2016 *)

a(n) = if(n == 0, 0, my(f = factor(n)); (5 * sigma(f, 2) - 3 * sigma(f)) / 2); \\ Amiram Eldar, Dec 29 2024

G.f.: Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).
Dirichlet g.f.: (5*zeta(s-2) - 3*zeta(s-1))*zeta(s)/2.
a(n) = Sum_{d|n} d*(5*d - 3)/2.
a(n) = (5*A001157(n) - 3*A000203(n))/2.
Sum_{k=1..n} a(k) ~ (5*zeta(3)/6) * n^3. - Amiram Eldar, Dec 29 2024

A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

Original entry on oeis.org

1, 6, 15, 36, 56, 111, 141, 240, 300, 446, 507, 791, 820, 1161, 1310, 1736, 1786, 2505, 2471, 3346, 3466, 4307, 4325, 5895, 5581, 7026, 7230, 8905, 8556, 11246, 10417, 13176, 13050, 15476, 15106, 19391, 17576, 21495, 21374, 25690, 23822, 30162, 27435, 33707, 32990, 37841, 35721

Offset: 1

Ilya Gutkovskiy, Oct 24 2018

Inverse Möbius transform of square pyramidal numbers (A000330).

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000203, A000330, A001157, A001158, A007437, A059358.

Cf. A135539.

a:=series(add(x^k*(1+x^k)/(1-x^k)^4,k=1..100),x=0,48): seq(coeff(a,x,n),n=1..47); # Paolo P. Lava, Apr 02 2019

nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]
Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]

a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 2) + sigma(f, 1)) / 6; \\ Amiram Eldar, Jan 03 2025

G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.
a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.
a(n) = Sum_{i=1..n} i^2*A135539(n,i). - Ridouane Oudra, Jul 22 2022
From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: zeta(s) * (2*zeta(s-3) + 3*zeta(s-2) + zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/12) * n^4. (End)

A116989 a(n) = b(A000579(n+6)) with b(n) = Sum{d|n} d(d+1)(d+2)(d+3)(d+4)*(d+5)/720.

Original entry on oeis.org

1, 925, 1135716, 593223373, 130220375812, 14195655302684, 893936543319276, 36397263567477054, 1025115791220794876, 21899052879460199956, 372805053916689840596, 5076066733212581886566, 57875038239259949679248

Offset: 0

Jonathan Vos Post, Apr 02 2006

Cf. A000579, A007437, A101289, A116963.

a(n) = sumdiv(binomial(n+6, 6), d, d*(d+1)*(d+2)*(d+3)*(d+4)*(d+5)/720) /* Georg Fischer, Aug 03 2025 */

Definition corrected by Georg Fischer, Aug 03 2025

A134544 A051731 * A002260.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 2, 2, 3, 4, 5, 4, 6, 6, 4, 5, 6, 2, 2, 3, 4, 5, 6, 7, 4, 6, 6, 8, 5, 6, 7, 8, 3, 4, 6, 4, 5, 6, 7, 8, 9, 4, 6, 6, 8, 10, 6, 7, 8, 9, 10

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = A007437: (1, 4, 7, 14, 16, 3, 29, 50, ...).

Left border = A000010.

			First few rows of the triangle:
  1;
  2,  2;
  2,  2,  3;
  3,  4,  3,  4;
  2,  2,  3,  4,  5;
  4,  6,  6,  4,  5,  6;
  2,  2,  3,  4,  5,  6,  7;
  4,  6,  6,  8,  5,  6,  7,  8;
  3,  4,  6,  4,  5,  6,  7,  8,  9;
  4,  6,  6,  8, 10,  6,  7,  8,  9, 10;
  ...

Cf. A054731, A007437, A000010.

A051731 * A002260 as infinite lower triangular matrices.

A134545 A051731 * A004736.

Original entry on oeis.org

1, 3, 1, 4, 2, 1, 7, 4, 2, 1, 6, 4, 3, 2, 1, 12, 8, 5, 3, 2, 1, 8, 6, 5, 4, 3, 2, 1, 15, 11, 8, 6, 4, 3, 2, 1, 13, 10, 8, 6, 5, 4, 3, 2, 1, 18, 14, 11, 9, 7, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = A007437: (1, 4, 7, 14, 16, 31, ...).

Left border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

			First few rows of the triangle:
   1;
   3,  1;
   4,  2, 1;
   7,  4, 2, 1;
   6,  4, 3, 2, 1;
  12,  8, 5, 3, 2, 1;
   8,  6, 5, 4, 3, 2, 1;
  15, 11, 8, 6, 4, 3, 2, 1;
  ...

Cf. A051731, A004736, A007436, A000203.

A051731 * A004736 as infinite lower triangular matrices.

A134839 Triangle, n-th row = first n terms in n-th row of an array formed by A051731 * A126988(transform).

Original entry on oeis.org

1, 1, 3, 1, 2, 4, 1, 3, 3, 7, 1, 2, 3, 4, 6, 1, 3, 4, 6, 5, 12, 1, 2, 3, 4, 5, 6, 8, 1, 3, 3, 7, 5, 9, 7, 15, 1, 2, 4, 4, 5, 8, 7, 8, 13, 1, 3, 3, 6, 6, 9, 7, 12, 9, 18

Offset: 1

Gary W. Adamson, Nov 12 2007

Right border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

Row sums = A007437: (1, 4, 7, 14, 16, 31, 29, 50, ...).

			First few rows of the array:
  1,  2,  3,  4,  5,  6,  7, ...
  1,  3,  3,  6,  5,  9,  7, ...
  1,  2,  4,  4,  5,  8,  7, ...
  1,  3,  3,  7,  5,  9,  7, ...
  1,  2,  3,  4,  6,  6,  7, ...
  1,  3,  4,  6,  5, 12,  7, ...
  ...
First few rows of the triangle:
  1;
  1,  3;
  1,  2,  4;
  1,  3,  3,  7;
  1,  2,  3,  4,  6;
  1,  3,  4,  6,  5, 12;
  1,  2,  3,  4,  5,  6,  8;
  ...

Cf. A126988, A051731, A000203, A007437.

Given the array formed by A051731 * A126988(transform), the triangle by rows = first n terms of n-th row of the array.

A275786 a(n) = Product_{d|n} T(d) where T(x) = x*(x+1)/2 = A000217(x) = x-th triangular number.

Original entry on oeis.org

1, 3, 6, 30, 15, 378, 28, 1080, 270, 2475, 66, 294840, 91, 8820, 10800, 146880, 153, 2908710, 190, 5197500, 38808, 50094, 276, 3184272000, 4875, 95823, 102060, 35809200, 435, 17401230000, 496, 77552640, 222156, 273105, 264600, 1511016670800, 703, 422370, 425880

Offset: 1

Jaroslav Krizek, Aug 09 2016

nonn

Conjecture: the sequence is injective (all terms of this sequence occur only once).

			a(4) = 30 because the divisors of 4 are: 1, 2 and 4; and T(1)*T(2)*T(4) = 1*3*10 = 30.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000217, A007437 (Sum_{d|n} T(d)).

[(&*[d*(d+1) div 2: d in Divisors(n)]): n in [1..100]]

f:= n -> convert(map(t -> t*(t+1)/2,numtheory:-divisors(n)),`*`):
map(f, [$1..100]); # Robert Israel, Aug 09 2016

t[n_]:=Divisors[n]*(Divisors[n]+1)/2;a[n_]:=Times@@t[n];Array[a,50] (* Ivan N. Ianakiev, Aug 15 2016 *)

a(p) = A000217(p) = p*(p+1)/2 for a prime p.

A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

Original entry on oeis.org

1, 5, 11, 24, 32, 61, 65, 109, 120, 172, 167, 279, 236, 343, 358, 470, 410, 630, 515, 762, 706, 865, 761, 1193, 933, 1216, 1174, 1497, 1220, 1850, 1397, 1959, 1762, 2098, 1882, 2739, 2000, 2629, 2470, 3188, 2462, 3614, 2711, 3723, 3438, 3871, 3245, 4939, 3594, 4749, 4246, 5214

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

Inverse Moebius transform of centered triangular numbers (A005448).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001157, A002117, A005448, A007437, A059358.

nmax = 52; CoefficientList[Series[Sum[x^k (1 - x^(3 k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[3 (DivisorSigma[2, n] - DivisorSigma[1, n])/2 + DivisorSigma[0, n], {n, 1, 52}]

a(n)={sumdiv(n, d, 3*d*(d-1)/2 + 1)} \\ Andrew Howroyd, Aug 14 2019

a(n)={3*(sigma(n,2) - sigma(n))/2 + numdiv(n)} \\ Andrew Howroyd, Aug 14 2019

G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^k/(1 - x^k).
a(n) = 3 * (sigma_2(n) - sigma_1(n))/2 + d(n).
From Amiram Eldar, Jan 02 2025: (Start)
Dirichlet g.f.: zeta(s) * (3 * zeta(s-2) - 3 * zeta(s-1) + 2 * zeta(s)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)

A326826 a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.

Original entry on oeis.org

1, 5, 8, 19, 17, 43, 30, 69, 60, 95, 68, 176, 93, 171, 166, 255, 155, 342, 192, 403, 303, 395, 278, 681, 358, 543, 490, 738, 437, 961, 498, 969, 709, 911, 720, 1476, 705, 1131, 978, 1603, 863, 1773, 948, 1732, 1440, 1643, 1130, 2634, 1284, 2110, 1648, 2391, 1433, 2882, 1706

Offset: 1

Ilya Gutkovskiy, Oct 20 2019

nonn

Inverse Moebius transform applied twice to triangular numbers (A000217).

Cf. A000005, A000203, A000217, A001157, A007429, A007433, A007437, A092346, A326828.

[(1/2)*&+[DivisorSigma(1,d)+DivisorSigma(2,d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 20 2019

with(numtheory):
a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019

Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ Michel Marcus, Oct 20 2019

G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j) / (1 - x^(i*j))^3.
G.f.: (1/2) * Sum_{i>=1} Sum_{j>=1} j * (j + 1) * x^(i*j) / (1 - x^(i*j)).
G.f.: (1/2) * Sum_{k>=1} (sigma_1(k) + sigma_2(k)) * x^k / (1 - x^k).
Dirichlet g.f.: zeta(s)^2 * (zeta(s-1) + zeta(s-2)) / 2.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * tau(n/d), where tau = A000005.
a(n) = Sum_{d|n} A007437(d).
Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Dec 11 2021

cp's OEIS Frontend

A366135 Expansion of Sum_{k>=1} k^3 * x^k/(1 - x^k)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A278947 Expansion of Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A116989 a(n) = b(A000579(n+6)) with b(n) = Sum{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)*(d+5)/720.

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A134544 A051731 * A002260.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A134545 A051731 * A004736.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A134839 Triangle, n-th row = first n terms in n-th row of an array formed by A051731 * A126988(transform).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A275786 a(n) = Product_{d|n} T(d) where T(x) = x*(x+1)/2 = A000217(x) = x-th triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

A278947 Expansion of Sum_{k>=1} (k(5k - 3)/2)*x^k/(1 - x^k).

A116989 a(n) = b(A000579(n+6)) with b(n) = Sum{d|n} d(d+1)(d+2)(d+3)(d+4)*(d+5)/720.