A013666 -id:A013666 - cp's OEIS frontend

A085968 Decimal expansion of the prime zeta function at 8.

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

			0.0040614053665178305605...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1971
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085967 (at 7), this sequence (at 8), A085969 (at 9).

Cf. A013666, A179645.

R := RealField(106);
PrimeZeta := func;
[0,0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(8,43)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[8*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n = 200]; While[s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 8], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p, 8) \\ Hugo Pfoertner, Feb 03 2020

P(8) = Sum_{p prime} 1/p^8 = Sum_{n>=1} mobius(n)*log(zeta(8*n))/n.

Equals Sum_{k>=1} 1/A179645(k). - Amiram Eldar, Jul 27 2020

A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 7th Bendersky constant.

			0.9899756533334170941753964830588692002082471514307453051285538624....

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)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), 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, A013666, A259072, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 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(7) = exp(H(7)*B(8)/8 - zeta'(-7)) = exp((B(8)/8)*(EulerGamma + log(2*Pi) - (zeta'(8)/zeta(8)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^8-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(8)/8 = -1/240 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266555 Decimal expansion of the generalized Glaisher-Kinkelin constant A(8).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 8th Bendersky constant.

			0.99171832163282219699954748276579333986785976057305079247...

G. C. Greubel, Table of n, a(n) for n = 0..2001

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)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), 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. A013666, A013667, A259073, A027641, A027642.

Exp[N[(BernoulliB[8]/4)*(Zeta[9]/Zeta[8]), 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(8) = -zeta'(-8) = (B(8)/4)*(zeta(9)/zeta(8)).
A(8) = exp(-8! * Zeta(9) / (2^9 * Pi^8)). - 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

A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 258, 6564, 66052, 390630, 1693512, 5764808, 16909320, 43066413, 100782540, 214358892, 433565328, 815730734, 1487320464, 2564095320, 4328785936, 6975757458, 11111134554, 16983563060, 25801892760, 37840199712, 55304594136, 78310985304, 110992776480

Offset: 0

Seiichi Manyama, Feb 05 2017

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

			a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), this sequence (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A013955, A013666, A196288.

terms = 25;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^8*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018

G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282101(n) - A013974(n))/720. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
G.f. Sum_{k>=1} k^8*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A351270 Sum of the 7th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 129, 2188, 129, 78126, 282252, 823544, 129, 2188, 10078254, 19487172, 282252, 62748518, 106237176, 170939688, 129, 410338674, 282252, 893871740, 10078254, 1801914272, 2513845188, 3404825448, 282252, 78126, 8094558822, 2188, 106237176, 17249876310, 22051219752, 27512614112

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^7 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 129; a(4) = Sum_{d|4} d^7 * mu(d)^2 = 1^7*1 + 2^7*1 + 4^7*0 = 129.

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

Cf. A008683 (mu), A013661, A013666.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), this sequence (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^7); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^7 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^7. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^7 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/(8*zeta(2)) = Pi^6/12600 = 0.0763007... . - Amiram Eldar, Nov 10 2022

A351604 a(n) = n^6 * Sum_{d^2|n} 1 / d^6.

Original entry on oeis.org

1, 64, 729, 4160, 15625, 46656, 117649, 266240, 532170, 1000000, 1771561, 3032640, 4826809, 7529536, 11390625, 17043456, 24137569, 34058880, 47045881, 65000000, 85766121, 113379904, 148035889, 194088960, 244156250, 308915776, 387951930, 489419840, 594823321, 729000000

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), this sequence (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013666.

f[p_, e_] := p^6*(p^(6*e) - p^(6*Floor[(e - 1)/2]))/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^6*sumdiv(n, d, if (issquare(d), 1/d^3)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^6*(p^(6*e) - p^(6*floor((e-1)/2)))/(p^6 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(8)/7 = Pi^8/66150 = 0.143439... . - Amiram Eldar, Nov 13 2022

A352035 Sum of the 7th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2188, 1, 1, 2188, 78126, 1, 2188, 1, 823544, 80313, 1, 1, 4785157, 1, 78126, 825731, 19487172, 1, 2188, 78126, 62748518, 4785157, 823544, 1, 170939688, 1, 1, 19489359, 410338674, 901669, 4785157, 1, 893871740, 62750705, 78126, 1, 1801914272, 1

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 78126; a(10) = Sum_{d|10, d<10, d odd} d^7 = 1^7 + 5^7 = 78126.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), this sequence (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013666, A321811.

f[2, e_] := 1; f[p_, e_] := (p^(7*e+7) - 1)/(p^7 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^7, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^7 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321811(n) - n^7*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)-1)/16 = 0.0002548347... . (End)

A352053 Sum of the 7th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 128, 2187, 16384, 78125, 280064, 823543, 2097152, 4785156, 10000128, 19487171, 35848192, 62748517, 105413632, 170939687, 268435456, 410338673, 612500096, 893871739, 1280016384, 1801914271, 2494358016, 3404825447, 4588568576, 6103593750, 8031810304, 10465138359

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^7 * Sum_{d|10, d<10, d odd} 1/d^7 = 10^7 * (1/1^7 + 1/5^7) = 10000128.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), this sequence (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013666, A321811.

A352053[n_]:=DivisorSum[n,1/#^7&,#A352053,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-7, n/2^IntegerExponent[n, 2]] * n^7 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^7 * sigma(n >> valuation(n, 2), -7) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^7 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^7 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321811(n) * A006519(n)^7 - A000035(n).

Sum_{k=1..n} a(k) = c * n^8 / 8, where c = 255*zeta(8)/256 = 1.000155179... . (End)

A009694 a(n) = Product_{i=0..7} floor((n+i)/8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 81920, 102400, 128000, 160000, 200000, 250000, 312500

Offset: 0

N. J. A. Sloane

For n >= 8, a(n) is the maximal product of eight positive integers with sum n. - Wesley Ivan Hurt, Jul 08 2022

A quasipolynomial of order 8 and degree 8. - Charles R Greathouse IV, Nov 06 2022

Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 7, -14, 7, 0, 0, 0, 0, 0, -21, 42, -21, 0, 0, 0, 0, 0, 35, -70, 35, 0, 0, 0, 0, 0, -35, 70, -35, 0, 0, 0, 0, 0, 21, -42, 21, 0, 0, 0, 0, 0, -7, 14, -7, 0, 0, 0, 0, 0, 1, -2, 1).

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), this sequence (k=8), A009714 (k=9), A354600 (k=10).

Cf. A001016, A013666.

Table[Product[Floor[(n+i)/8],{i,0,7}],{n,0,40}] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = prod(i=0, 7, (n+i)\8); \\ Michel Marcus, Jul 14 2022

a(8*n) = n^8 (A001016). - Bernard Schott, Nov 06 2022
a(n) = n^8/8^8 + O(n^6). - Charles R Greathouse IV, Nov 06 2022
Sum_{n>=8} 1/a(n) = 1 + zeta(8). - Amiram Eldar, Jan 10 2023

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cp's OEIS Frontend

A085968 Decimal expansion of the prime zeta function at 8.

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A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

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A266555 Decimal expansion of the generalized Glaisher-Kinkelin constant A(8).

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

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A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A351270 Sum of the 7th powers of the squarefree divisors of n.

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A351604 a(n) = n^6 * Sum_{d^2|n} 1 / d^6.

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A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.