A283078 -id:A283078 - cp's OEIS frontend

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

1, 8, 44, 192, 726, 2464, 7704, 22527, 62329, 164516, 416948, 1019690, 2416246, 5565864, 12498215, 27421815, 58903768, 124088548, 256749822, 522450250, 1046735092, 2066948472, 4026431543, 7743987036, 14715788745, 27648250012, 51390298666, 94550761844

Offset: 0

Seiichi Manyama, Feb 28 2017

nonn

			G.f.: A(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + ...
log(A(x)) = 8*x + 24*x^2/2 + 32*x^3/3 + 56*x^4/4 + 48*x^5/5 + 96*x^6/6 + 57*x^7/7 + 120*x^8/8 + ... + sigma(7*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282942 (Product_{n>=1} (1 - x^n)^8/(1 - x^(7*n))), A283078 (sigma(7*n)).

Cf. exp( Sum_{n>=1} sigma(k*n)*x^n/n ): A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), this sequence (k=7), A283120 (k=8), A283121 (k=9).

G.f.: exp( Sum_{n>=1} sigma(7*n)*x^n/n ).
a(n) = (1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 3025 * exp(sqrt(110*n/21)*Pi) / (28224*sqrt(14)*n^(5/2)). - Vaclav Kotesovec, Mar 20 2017

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 400, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8

Offset: 1

Labos Elemer, Nov 04 2003

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

Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).

Cf. A000203, A038712, A080278, A214411, A268354, A283078.

Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(7*n)/sigma(n)); \\ Amiram Eldar, Mar 22 2024

From Amiram Eldar, Mar 22 2024: (Start)
a(n) = numerator(A283078(n)/A000203(n)).
a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.
Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

A319526 Square array read by antidiagonals upwards: T(n,k) = sigma(n*k), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 13, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 31, 24, 28, 8, 15, 24, 39, 42, 42, 39, 24, 15, 13, 31, 32, 60, 31, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 40, 63, 48, 91, 48, 63, 40, 42, 12, 28, 36, 72, 91, 90, 96, 96, 90, 91, 72, 36, 28

Offset: 1

Omar E. Pol, Sep 25 2018

			The corner of the square array begins:
A000203:    1,   3,   4,   7,   6,  12,   8,  15,  13,  18,  12,  28, ...
A062731:    3,   7,  12,  15,  18,  28,  24,  31,  39,  42,  36,  60, ...
A144613:    4,  12,  13,  28,  24,  39,  32,  60,  40,  72,  48,  91, ...
A193553:    7,  15,  28,  31,  42,  60,  56,  63,  91,  90,  84, 124, ...
A283118:    6,  18,  24,  42,  31,  72,  48,  90,  78,  93,  72, 168, ...
A224613:   12,  28,  39,  60,  72,  91,  96, 124, 120, 168, 144, 195, ...
A283078:    8,  24,  32,  56,  48,  96,  57, 120, 104, 144,  96, 224, ...
A283122:   15,  31,  60,  63,  90, 124, 120, 127, 195, 186, 180, 252, ...
A283123:   13,  39,  40,  91,  78, 120, 104, 195, 121, 234, 156, 280, ...
...

Index entries for sequences related to sigma(n)

First 9 rows (also first 9 columns) are A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.
Main diagonal gives A065764.
Cf. A000203, A003991, A216626, A319073.

Table[DivisorSigma[1, # k] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

T(n,k) = A000203(n*k).

T(n,k) = A000203(A003991(n,k)).

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Original entry on oeis.org

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

Table[Sum[DivisorSigma[1, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[1, n^2] + 2*Sum[DivisorSigma[1, j*n], {j, 1, n - 1}], {n, 2, 50}]]

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

cp's OEIS Frontend

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319526 Square array read by antidiagonals upwards: T(n,k) = sigma(n*k), n >= 1, k >= 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula