A013666 -id:A013666 - cp's OEIS frontend

A293904 Decimal expansion of zeta(21).

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

Offset: 1

Frank Ellermann, Oct 19 2017

Web searches find 1.0000004769329867878 in Python tools. Simon Plouffe published 1000 digits for zeta(9) up to zeta(2051) many years ago.

			1.000000476932986787806...

Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each.
Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each. [Cached copy, with permission]
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each. (gzipped file)
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, July 21, 1998.

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678.

RealDigits[Zeta[21], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

A069092 Jordan function J_7(n).

Original entry on oeis.org

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]
Cf. A013666.

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1, 100, print1(sumdiv(n, d, d^7*moebius(n/d)), ", "))

a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

Original entry on oeis.org

1, 127, 2188, 16255, 78126, 277876, 823544, 2080639, 4785157, 9922002, 19487172, 35565940, 62748518, 104590088, 170939688, 266321791, 410338674, 607714939, 893871740, 1269938130, 1801914272, 2474870844, 3404825448, 4552438132, 6103593751, 7969061786, 10465138360, 13386707720, 17249876310

Offset: 1

N. J. A. Sloane, Nov 23 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_{k>=1} k^b*x^k/(1 + x^k): A000593 (b=1), A078306 (b=2), A078307 (b=3), A284900 (b=4), A284926 (b=5), A284927 (b=6), this sequence (b=7), A321553 (b=8), A321554 (b=9), A321555 (b=10), A321556 (b=11), A321557 (b=12).
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013666.

f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := (63*2^(7*e + 1) + 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321552(n)=sumdiv(n, d, (-1)^(n\d-1)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (63*2^(7*e+1)+1)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 127*zeta(8)/1024 = 0.124529... . (End)

A096961 a(n) = Sum_{0

Original entry on oeis.org

1, 128, 2188, 16384, 78126, 280064, 823544, 2097152, 4785157, 10000128, 19487172, 35848192, 62748518, 105413632, 170939688, 268435456, 410338674, 612500096, 893871740, 1280016384, 1801914272, 2494358016, 3404825448

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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.

Cf. A007331, A008410, A013666, A096960, A096962, A096963.

A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* Michael Somos, Nov 30 2014 */

a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))};

ModularForms( Gamma0(2), 8, prec=24).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - Michael Somos, Jan 09 2015
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 516, 19692, 264208, 1953150, 10161072, 40353656, 135274560, 387597717, 1007825400, 2357947812, 5202783936, 10604499542, 20822486496, 38461429800, 69260574976, 118587876786, 200000421972, 322687698140, 516037855200, 794644193952, 1216701070992

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).
Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).
Cf. A013666.

Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* Indranil Ghosh, Mar 12 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013955(n) for n > 0.
a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.
Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)
G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

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 *)

A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 121, 2161, 15481, 78001, 261481, 823201, 1981561, 4726081, 9438121, 19485841, 33454441, 62746321, 99607321, 168560161, 253639801, 410333761, 571855801, 893864881, 1207533481, 1778937361, 2357786761, 3404813281, 4282153321, 6093828001, 7592304841, 10335939121

Offset: 1

Seiichi Manyama, May 18 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A068970, A084220, A372950.
Cf. A008683, A013955.
Cf. A013663, A013666.
Cf. A350156, A372962.

f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A308637 Decimal expansion of Pi^3/Zeta(3).

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

-----+---------------------------------
   n |  Zeta(n)
-----+---------------------------------
   2 |     Pi^2  / 6         = A013661.
   3 |     Pi^3  / 25.79...  = A002117.
   4 |     Pi^4  / 90        = A013662.
   5 |     Pi^5  / A309926   = A013663.
   6 |     Pi^6  / 945       = A013664.
   7 |     Pi^7  / A309927   = A013665.
   8 |     Pi^8  / 9450      = A013666.
   9 |     Pi^9  / A309928   = A013667.
  10 |     Pi^10 / 93555     = A013668.
  11 |     Pi^11 / A309929   = A013669.
  12 | 691*Pi^12 / 638512875 = A013670.
...
Cf. A002432, A091925, A276120 (Zeta(3)/Pi^3).

RealDigits[Pi^3/Zeta[3], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^3/zeta(3)
```

Pi^3/Zeta(3) = A091925/A002117.

More terms from Amiram Eldar, Aug 24 2019

cp's OEIS Frontend

A293904 Decimal expansion of zeta(21).

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A069092 Jordan function J_7(n).

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A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

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A096961 a(n) = Sum_{0

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A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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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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