A013665 -id:A013665 - cp's OEIS frontend

A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

1, 1, 32, 275, 1763, 12421, 85808, 561074, 3535678, 21815897, 131733641, 778099521, 4505634324, 25635135074, 143507764032, 791243636305, 4300983535471, 23070300486656, 122213931799869, 639848848696540, 3312824859756453, 16972058378914997, 86082216143323410

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

In general, for m > 0, if g.f. = Product_{k>=1} (1+x^k)^(k^m), then a(n) ~ 2^(zeta(-m)) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2))) / (sqrt(2*Pi*(m+2)) * n^((m+3)/(2*m+4))).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A026007 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).

Column k=5 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^6, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5),{k,1,nmax}],{x,0,nmax}],x]

m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ (5*zeta(7))^(1/14) * 3^(2/7) * exp(zeta(7)^(1/7) * 2^(-9/7) * 3^(-3/7) * 5^(1/7) * 7^(8/7) * n^(6/7)) / (2^(163/252) * 7^(3/7) * sqrt(Pi) * n^(4/7)), where zeta(7) = A013665.

A284927 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.

Original entry on oeis.org

1, 63, 730, 4031, 15626, 45990, 117650, 257983, 532171, 984438, 1771562, 2942630, 4826810, 7411950, 11406980, 16510911, 24137570, 33526773, 47045882, 62988406, 85884500, 111608406, 148035890, 188327590, 244156251, 304089030, 387952660, 474247150, 594823322

Offset: 1

Seiichi Manyama, Apr 06 2017

Multiplicative because this sequence is the Dirichlet convolution of A001014 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), A284926 (k=5), this sequence (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A001014, A013665, A062157.

Table[Sum[(-1)^(n/d + 1)*d^6, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 06 2017 *)
f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := (31*2^(6*e + 1) + 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^6); \\ Indranil Ghosh, Apr 06 2017

from sympy import divisors
print([sum([(-1)**(n//d + 1)*d**6 for d in divisors(n)]) for n in range(1, 51)]) # Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (31*2^(6*e+1)+1)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = 9*zeta(7)/64 = 0.141799... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A344303 Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 128, 1094, 8256, 19532, 140032, 137258, 528448, 797891, 2500096, 1948718, 9032064, 5229044, 17569024, 21368008, 33820736, 25646168, 102130048, 49659542, 161256192, 150160252, 249435904, 154764794, 578122112, 305191407, 669317632, 581662904, 1133202048

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160897.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013665, A060648, A064969, A280184, A344219, A344302, A344304, A344305, A344306.

f[p_, e_] := 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160897(n) = sumdiv(n, d, moebius(n/d)*d^7)/eulerphi(n);
a(n) = sumdiv(n, d, a160897(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_7|n} phi(x_1)*phi(x_2)* ... *phi(x_7)/phi(lcm(x_1, x_2, ..., x_7)).
If p is prime, a(p) = 1 + (p^7 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)).
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)/7) * Product_{p prime} ((1-1/p^6)/(p^2*(1-1/p))) = 0.2784611791... . (End)

A275710 Decimal expansion of the Dirichlet eta function at 7.

Original entry on oeis.org

9, 9, 2, 5, 9, 3, 8, 1, 9, 9, 2, 2, 8, 3, 0, 2, 8, 2, 6, 7, 0, 4, 2, 5, 7, 1, 3, 1, 3, 3, 3, 9, 3, 6, 8, 5, 2, 3, 1, 1, 1, 5, 6, 9, 2, 4, 3, 1, 4, 0, 6, 8, 5, 1, 6, 2, 9, 5, 1, 3, 0, 8, 7, 5, 6, 2, 6, 7, 0, 2, 0, 5, 2, 1, 8, 6, 4, 7, 0, 5, 1, 9, 8, 1, 3, 1, 4, 2, 0, 3, 7, 7, 4, 5, 7, 2, 3, 9, 7, 0

Offset: 0

Terry D. Grant, Aug 06 2016

			0.99259381992283028267...

Cf. A002162 (value at 1), A013665, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275703 (value at 6), A334668, A334669, A347150, A347059.

RealDigits[63 Zeta[7]/64, 10, 100] [[1]]

-polylog(7, -1) \\ Michel Marcus, Aug 20 2021

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^7
print(s) # Terry D. Grant, Aug 06 2016

eta(7) = 63*zeta(7)/64 = (63*A013665)/64.
eta(7) = Lim_{n -> infinity} A334668(n)/A334669(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{k>=1} (-1)^(k+1) / k^7. - Sean A. Irvine, Aug 19 2021

A347217 Decimal expansion of Sum_{k=2..7} zeta(k).

Original entry on oeis.org

6, 9, 9, 1, 9, 3, 4, 2, 9, 8, 2, 2, 8, 7, 0, 0, 8, 0, 6, 2, 7, 3, 8, 3, 7, 9, 9, 0, 7, 9, 6, 3, 9, 4, 5, 3, 8, 3, 1, 7, 4, 4, 9, 1, 1, 5, 5, 6, 6, 0, 2, 9, 3, 0, 7, 4, 4, 4, 6, 9, 0, 4, 7, 8, 6, 3, 5, 7, 0, 3, 5, 4, 3, 3, 8, 2, 0, 7, 7, 9, 2, 8, 3, 6, 5, 9, 0

Offset: 1

Sean A. Irvine, Aug 24 2021

			6.9919342982287008062738379907963945383174491155660...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665.

Cf. A347213, A347214, A347215, A347216, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 7]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

Original entry on oeis.org

7, 9, 9, 6, 0, 1, 1, 6, 5, 4, 4, 2, 6, 6, 4, 5, 1, 4, 5, 6, 5, 2, 5, 2, 3, 2, 2, 9, 3, 0, 5, 0, 4, 7, 0, 0, 3, 5, 7, 6, 4, 0, 9, 9, 0, 6, 2, 1, 5, 8, 7, 9, 3, 2, 7, 7, 7, 3, 8, 0, 0, 6, 8, 1, 2, 8, 8, 2, 8, 6, 4, 9, 5, 9, 5, 6, 8, 2, 6, 7, 4, 2, 7, 6, 1, 1, 9

Offset: 1

Sean A. Irvine, Aug 24 2021

			7.99601165442664514565252322930504700357640...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666.

Cf. A347213, A347214, A347215, A347216, A347217, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 8]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

Equals A347217 + A013666. - R. J. Mathar, May 27 2024

A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

Original entry on oeis.org

8, 9, 9, 8, 0, 2, 0, 0, 4, 7, 2, 5, 2, 7, 2, 7, 3, 6, 0, 0, 7, 0, 3, 7, 5, 9, 9, 8, 5, 3, 7, 4, 5, 9, 0, 6, 4, 0, 6, 2, 0, 1, 5, 7, 5, 7, 6, 1, 0, 7, 6, 8, 0, 8, 4, 3, 2, 2, 3, 9, 7, 2, 9, 7, 1, 9, 8, 0, 7, 1, 5, 4, 9, 3, 4, 7, 0, 8, 5, 1, 4, 1, 7, 1, 1, 5, 8

Offset: 1

Sean A. Irvine, Aug 24 2021

			8.9980200472527273600703759985374590640620...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347220.

RealDigits[Total[Zeta[Range[2, 9]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

Original entry on oeis.org

9, 9, 9, 9, 0, 1, 4, 6, 2, 2, 3, 8, 0, 5, 4, 5, 4, 4, 5, 4, 0, 7, 5, 2, 1, 9, 5, 7, 4, 3, 7, 7, 7, 8, 0, 8, 1, 0, 6, 8, 0, 3, 5, 2, 8, 9, 1, 7, 5, 2, 4, 5, 6, 0, 1, 5, 8, 0, 1, 8, 6, 2, 9, 1, 8, 3, 4, 3, 6, 3, 0, 1, 4, 4, 9, 8, 9, 9, 8, 0, 9, 6, 1, 0, 8, 6, 3

Offset: 1

Sean A. Irvine, Aug 24 2021

			9.9990146223805454454075219574377780810680...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667, A013668.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347219.

RealDigits[Total[Zeta[Range[2, 10]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A023874 Expansion of Product_{k>=1} (1 - x^k)^(-k^5).

Original entry on oeis.org

1, 1, 33, 276, 1828, 12729, 88903, 582846, 3690325, 22864592, 138658796, 822374485, 4781447342, 27314310586, 153519181630, 849786024496, 4637270263913, 24970548655999, 132788838463944, 697863705334941, 3626864249759775, 18650694625385462, 94948991121030892

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2338 (first 601 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Column k=5 of A144048.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^5, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2012

max = 22; Series[ Product[1/(1 - x^k)^k^5, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^5)) \\ G. C. Greubel, Oct 30 2018

a(n) ~ 3^(127/882) * (5*Zeta(7))^(127/1764) * exp(7 * n^(6/7) * (5*Zeta(7))^(1/7) / (2^(3/7) * 3^(5/7)) + Zeta'(-5)) / (2^(187/882) * n^(1009/1764) * sqrt(7*Pi)), where Zeta(7) = A013665 = 1.008349277381922826..., Zeta'(-5) = ((137/60 - gamma - log(2*Pi))/42 + 45*Zeta'(6) / (2*Pi^6))/6 = -0.0005729859801986352... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_6(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_6(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A113852 Numbers whose prime factors are raised to the seventh power.

Original entry on oeis.org

128, 2187, 78125, 279936, 823543, 10000000, 19487171, 62748517, 105413504, 170859375, 410338673, 893871739, 1801088541, 2494357888, 3404825447, 8031810176, 17249876309, 21870000000, 27512614111, 42618442977, 52523350144, 64339296875, 94931877133, 114415582592

Offset: 1

Cino Hilliard, Jan 25 2006

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

Proper subset of A001015.
Nonunit terms of A329332 column 7 in ascending order.
Cf. A005117, A013665, A013672.

Select[Range@34^7, Union[Last /@ FactorInteger@# ] == {7} &] (* Robert G. Wilson v, Jan 26 2006 *)
Select[Range[2, 34], SquareFreeQ]^7 (* Amiram Eldar, Oct 13 2020 *)

allpwrfact(n,p) = /* All prime factors are raised to the power p */ { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }

from math import isqrt
from sympy import mobius
def A113852(n):
    def f(x): return int(n+1-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**7 # Chai Wah Wu, Feb 25 2025

From Amiram Eldar, Oct 13 2020: (Start)
a(n) = A005117(n+1)^7.
Sum_{n>=1} 1/a(n) = zeta(7)/zeta(14) - 1. (End)

More terms from Robert G. Wilson v, Jan 26 2006

cp's OEIS Frontend

A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

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A284927 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.

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A344303 Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

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A275710 Decimal expansion of the Dirichlet eta function at 7.

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A347217 Decimal expansion of Sum_{k=2..7} zeta(k).

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A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

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A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

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A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

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