A013670 -id:A013670 - cp's OEIS frontend

A266558 Decimal expansion of the generalized Glaisher-Kinkelin constant A(11).

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 11th Bendersky constant.

			0.950331248453288866514233841015331271597566403456173040861088881...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013670, A266262, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[12]/12)*(EulerGamma + Log[2*Pi] - Zeta'[12]/Zeta[12]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(11) = exp(H(11)*B(12)/12 - zeta'(-11)) = exp((B(12)/12)*(EulerGamma + log(2*Pi) - (zeta'(12)/zeta(12)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^12-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(12)/12 = -691/32760 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266559 Decimal expansion of the generalized Glaisher-Kinkelin constant A(12).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 12th Bendersky constant.

			0.9386894455960125851529657813206767183332587685218350098663907...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A013670, A013671, A266263, A027641, A027642.

RealDigits[Exp[N[(BernoulliB[12]/4)*(Zeta[13]/Zeta[12]),200]]][[1]] (* Program amended by Harvey P. Dale, Aug 16 2021 *)

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(12) = exp(-zeta'(-12)) = exp((B(12)/4)*(zeta(13)/zeta(12))).
A(12) = exp(-12! * Zeta(13) / (2^13 * Pi^12)). - Vaclav Kotesovec, Jan 01 2016

A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.0866629762657094129329746026249997547771718667980916672...
= -3 + Pi^4/90 + Pi^8/9450 + 691*Pi^12/638512875 + ...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 265.

V. S. Adamchik, H. M. Srivastava, Some series of Zeta and related functions, Analysis (Munich) 18 (2) (1998) 131-144, eq. (2.25).
Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A013662, A013666, A013670, A175316, A256920, A339529, A339530, A339604 (3k-1).

Join[{0}, RealDigits[7/8 - (Pi/4)*Coth[Pi], 10, 104] // First]

Equals 7/8 - (Pi/4)*coth(Pi).
Equals Sum_{n>=2} 1/(n^4 - 1). - Vaclav Kotesovec, Dec 08 2020
Equals (1/2)* Sum_{n>=2} 1/(n^2-1) - (1/2)* Sum_{n>=2} 1/(n^2+1) = (3/4 - A100554)/2. - R. J. Mathar, Jan 22 2021

A351608 a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

Original entry on oeis.org

1, 1024, 59049, 1049600, 9765625, 60466176, 282475249, 1074790400, 3486843450, 10000000000, 25937424601, 61977830400, 137858491849, 289254654976, 576650390625, 1100586418176, 2015993900449, 3570527692800, 6131066257801, 10250000000000, 16679880978201

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

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

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), this sequence (k=10).

Cf. A013670.

f[p_, e_] := p^10*(p^(10*e) - p^(10*Floor[(e - 1)/2]))/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^10*sumdiv(n, d, if (issquare(d), 1/d^5)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^10*(p^(10*e) - p^(10*floor((e-1)/2)))/(p^10 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(12)/11 = 691*Pi^12/7023641625 = 0.090931... . - Amiram Eldar, Nov 13 2022

A293904 Decimal expansion of zeta(21).

Original entry on oeis.org

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 *)

A096963 a(n) = Sum {0

Original entry on oeis.org

1, 2048, 177148, 4194304, 48828126, 362799104, 1977326744, 8589934592, 31381236757, 100000002048, 285311670612, 743012564992, 1792160394038, 4049565171712, 8649804864648, 17592186044416, 34271896307634

Offset: 1

Ralf Stephan, Jul 18 2004

This is the member k=11 of the k-family sigma^#_k(n) := Sum {0

This notation appears in the Ono et al. link, Theorem 5 (with k=3, see A007331) and Theorem 8 (with k=11). - Wolfdieter Lang, Jan 13 2017

			G.f. = q + 2048*q^2 + 177148*q^3 + 4194304*q^4 + 48828126*q^5 + ...

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).
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94, Case 24, Theorem 8.
Index entries for sequences mentioned by Glaisher.

Cf. A007331, A002131, A013670, A096960, A096961, A096962.

A := Basis( ModularForms( Gamma0(2), 12), 18); A[2] + 2048*A[3] + 177148*A[4]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^11 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Nov 30 2014 *)
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u4 = QPochhammer[ q^4]^8}, q (u1^4 + 2072 q u4 u1^3 + 210048 q^2 u4^2 u1^2 + 5660672 q^3 u4^3 u1 + 45285376 q^4 u4^4) / u1 ], {q, 0, n}]; (* Michael Somos, Nov 30 2014 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^11))}; /* Michael Somos, Nov 30 2014 */

ModularForms( Gamma0(2), 12, prec=18).3; # Michael Somos, Nov 30 2014

G.f.: Sum_{n>0} n^11 * x^n / (1 - x^(2*n)).

a(n) = Sum {0

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(11*e) and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^12, where c = 1365*zeta(12)/16384 = 691*Pi^12/7664025600 = 0.0833334904... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-11)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A321556 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^11.

Original entry on oeis.org

1, 2047, 177148, 4192255, 48828126, 362621956, 1977326744, 8585738239, 31381236757, 99951173922, 285311670612, 742649588740, 1792160394038, 4047587844968, 8649804864648, 17583591913471, 34271896307634, 64237391641579

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.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Cf. A013670.

f[p_, e_] := (p^(11*e + 11) - 1)/(p^11 - 1); f[2, e_] := (1023*2^(11*e + 1) + 1)/2047; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321556(n)=sumdiv(n, d, (-1)^(n\d-1)*d^11), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (1023*2^(11*e+1)+1)/2047, and a(p^e) = (p^(11*e+11) - 1)/(p^11 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = 2047*zeta(12)/24576 = 0.0833131... . (End)

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

A056551 Smallest cube divisible by n divided by largest cube which divides n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 1, 27, 1000, 1331, 216, 2197, 2744, 3375, 8, 4913, 216, 6859, 1000, 9261, 10648, 12167, 27, 125, 17576, 1, 2744, 24389, 27000, 29791, 8, 35937, 39304, 42875, 216, 50653, 54872, 59319, 125, 68921, 74088, 79507, 10648

Offset: 1

Henry Bottomley, Jun 25 2000

			a(16) = 8 since smallest cube divisible by 16 is 64 and smallest cube which divides 16 is 8 and 64/8 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000189, A000578, A008834, A019555, A048798, A050985, A053149, A053150, A056552.

Cf. A002117, A013670, A330596.

f[p_, e_] := p^If[Divisible[e, 3], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^3; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%3, f[i,1], 1))^3; } \\ Amiram Eldar, Oct 28 2022

a(n) = A053149(n)/A008834(n) = A048798(n)*A050985(n) = A056552(n)^3.
From Amiram Eldar, Oct 28 2022: (Start)
Multiplicative with a(p^e) = 1 if e is divisible by 3, and a(p^e) = p^3 otherwise.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(12)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013670 * A330596 / (4*A002117) = 0.1557163105... . (End)
Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(2*s-3)). - Amiram Eldar, Sep 16 2023

A160972 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 13.

Original entry on oeis.org

1, 4095, 265720, 8386560, 61035156, 1088123400, 2306881200, 17175674880, 47071500840, 249938963820, 313842837672, 2228476723200, 1941507093540, 9446678514000, 16218261652320, 35175782154240, 36413889826860, 192757795939800, 122961939948120, 511874997903360, 612984472464000

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^12 such that the quotient group Z^12 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
Index to Jordan function ratios J_k/J_1.

Column 12 of A263950.

Cf. A000010, A000203, A008683, A013669, A013670.

b = 13; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(11*e - 11) * (p^12-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^11, d, if(ispower(d, 12), moebius(sqrtnint(d, 12))*sigma(n^11/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^12 - 1)*f[i,1]^(11*f[i,2] - 11)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_12(n)/J_1(n) where J_12 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(11e-11) * (p^12-1) / (p-1).
For squarefree n, a(n) = A000203(n^11). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^12, where c = (1/12) * Product_{p prime} (1 + (p^11-1)/((p-1)*p^12)) = 0.1619398772... .
Sum_{k>=1} 1/a(k) = zeta(11)*zeta(12) * Product_{p prime} (1 - 2/p^12 + 1/p^23) = 1.0002481006668... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^12). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

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cp's OEIS Frontend

A266558 Decimal expansion of the generalized Glaisher-Kinkelin constant A(11).

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A266559 Decimal expansion of the generalized Glaisher-Kinkelin constant A(12).

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A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

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A351608 a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

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A293904 Decimal expansion of zeta(21).

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A096963 a(n) = Sum {0

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A321556 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^11.

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

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