A013663 -id:A013663 - cp's OEIS frontend

A082544 Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).

1, 31, 241, 991, 3091, 7501, 16531, 31711, 57781, 96601, 157651, 240031, 362491, 519961, 739201, 1012441, 1383721, 1822711, 2409241, 3091441, 3966301, 4974751, 6257461, 7680781, 9481681, 11474941, 13916191, 16610371, 19911151, 23435191

Offset: 1

Benoit Cloitre, May 11 2003

nonn

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Column k=5 of A344527.
Cf. A018805 (pairs), A071778 (triples), A082540 (quadruples), A343978.
Cf. A015650.

a(n)=sum(k=1,n,moebius(k)*floor(n/k)^5)

from functools import lru_cache
@lru_cache(maxsize=None)
def A082544(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082544(k1)
        j, k1 = j2, n//j2
    return n*(n**4-1)-c+j # Chai Wah Wu, Mar 29 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^5; a(n) is asymptotic to c*n^5 with c=0.9643....
Lim_{n->infinity} a(n)/n^5 = 1/zeta(5) = A343308. - Karl-Heinz Hofmann, Apr 11 2021
Lim_{n->infinity} n^5/a(n) =   zeta(5) = A013663. - Karl-Heinz Hofmann, Apr 11 2021
a(n) = n^5 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Original entry on oeis.org

1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136685.

Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A255050 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).

Original entry on oeis.org

1, 4, 20, 80, 305, 1072, 3622, 11676, 36450, 110240, 324936, 935076, 2635338, 7285560, 19795370, 52930360, 139462956, 362471020, 930186694, 2358867240, 5915606398, 14680528648, 36073675792, 87816701332, 211891552280, 506981067168, 1203337174120, 2834401172172

Offset: 0

Vaclav Kotesovec, Mar 08 2015

nonn

Number of partitions of n unlabeled objects of 4 colors. - Peter Dolland, Feb 20 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio

Column k=4 of A075196.

Cf. A000292, A005380, A217093, A255052.

with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*binomial(d+3, 3), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50); # after Alois P. Heinz

nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+3,3],{j,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{j>=1} 1/(1-x^j)^C(j+3,3).
a(n) ~ Zeta(5)^(829/3600) * exp(11/72 - Zeta(3)/(4*Pi^2) + Zeta'(-3)/6 - 121*Zeta(3)^2 / (360*Zeta(5)) - Pi^6/(1800*Zeta(5)) + 11*Pi^8*Zeta(3) / (108000*Zeta(5)^2) - Pi^16/(194400000*Zeta(5)^3) + Pi^2 * n^(1/5)/ (6*2^(2/5) * Zeta(5)^(1/5)) - 11*Pi^4 * Zeta(3) * n^(1/5) / (900*2^(2/5)*Zeta(5)^(6/5)) + Pi^12 * n^(1/5) / (1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Zeta(3) * n^(2/5) / (6*2^(4/5) * Zeta(5)^(2/5)) - Pi^8 * n^(2/5) / (9000 * 2^(4/5) * Zeta(5)^(7/5)) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * Zeta(5)^(3/5)) + 5 * Zeta(5)^(1/5) * n^(4/5) / 2^(8/5)) / (A^(11/6) * 2^(971/1800) * 5^(1/2) * Pi * n^(2629/3600)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .
EULER transform of 1, 4, 10, 20, 35, 56, 84, ... (= A000292(n+1)). - Peter Dolland, Feb 20 2025

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

Also zetamult(2, 2, 1). - Charles R Greathouse IV, Jan 04 2017

			0.2288103976033537597687461489416887919325093427198821602294071...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Jonathan Borwein and Roland Girgensohn, Evaluation of triple Euler Sums, Elec. Jour. of Comb., Vol. 3, Issue 1, 1996. Article R23 (see page 21).
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First

zetamult([3,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult([2,2,1]) \\ Charles R Greathouse IV, Jan 04 2017

Equals Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).

A267316 Decimal expansion of the Dirichlet eta function at 5.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

			1/1^5 - 1/2^5 + 1/3^5 - 1/4^5 + 1/5^5 - 1/6^5 + ... = 0.972119770446909305935655143553469532553513362...

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function

Cf. A002162 (value at 1), A013663, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A136676, A334604.

RealDigits[(15 Zeta[5])/16, 10, 120][[1]]

15*zeta(5)/16 \\ Michel Marcus, Feb 01 2016

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s += -((-1)^i/((i)^5))
print(s) # Terry D. Grant, Aug 05 2016

Equals Sum_{k > 0} (-1)^(k+1)/k^5 = (15*zeta(5))/16.

Equals Lim_{n -> infinity} A136676(n)/A334604(n). - Petros Hadjicostas, May 07 2020

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 *)

A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 32, 122, 528, 782, 3904, 2802, 8464, 9923, 25024, 16106, 64416, 30942, 89664, 95404, 135440, 88742, 317536, 137562, 412896, 341844, 515392, 292562, 1032608, 488907, 990144, 803804, 1479456, 732542, 3052928, 954306, 2167056, 1964932, 2839744, 2191164, 5239344, 1926222

Offset: 1

Seiichi Manyama, May 12 2021

Inverse Moebius transform of A160893.

Seiichi Manyama, 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, A013663, A060648, A064969, A160893, A280184.

f[p_, e_] := 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, eulerphi(i)*eulerphi(j)*eulerphi(k)*eulerphi(l)*eulerphi(m)/eulerphi(lcm([i, j, k, l, m])))))));

a160893(n) = sumdiv(n, d, moebius(n/d)*d^5)/eulerphi(n);
a(n) = sumdiv(n, d, a160893(d));

a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} phi(x_1)*phi(x_2)*phi(x_3)*phi(x_4)*phi(x_5)/phi(lcm(x_1, x_2, x_3, x_4, x_5)).
If p is prime, a(p) = 1 + (p^5 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)).
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3939461744... . (End)

A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

Original entry on oeis.org

1, 2, 34, 228, 1414, 8872, 52876, 301136, 1662614, 8929406, 46738920, 239036116, 1197187780, 5882369976, 28397283056, 134864166352, 630819797174, 2908948327780, 13236421303742, 59477002686404, 264104800719672, 1159649708139680, 5037895127964316

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023873 and A248883. - Vaclav Kotesovec, Aug 19 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
If m is even and m >= 2, then can be simplified as: a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + (-1)^(m/2) * Gamma(m+1) * Zeta(m+1) / (2^(m+1) * Pi^m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^16/(1-x^2)^16 * (1+x^3)^81/(1-x^3)^81 *...
Also, A(x) = Euler transform of [2,31,162,496,1250,2511,4802,7936,...]:
A(x) = 1/((1-x)^2*(1-x^2)^31*(1-x^3)^162*(1-x^4)^496*(1-x^5)^1250*(1-x^6)^2511*...).

Seiichi Manyama, Table of n, a(n) for n = 0..2916 (terms 0..1000 from Vaclav Kotesovec)
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. 23.

Cf. A015128 (m=0), A156616 (m=1), A206622 (m=2), A206623 (m=3), A001160 (sigma_5).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^4)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 5)-sigma(m, 5))/16*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+31*x+152*x^2+341*x^3+460*x^4+341*x^5+152*x^6+31*x^7+2*x^8)/(1-x^2+x*O(x^n))^5); polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_5(2*n) - sigma_5(n))/16 * x^n/n ), where sigma_5(n) is the sum of 5th powers of divisors of n (A001160).
Inverse Euler transform has g.f.: x*(2 + 31*x + 152*x^2 + 341*x^3 + 460*x^4 + 341*x^5 + 152*x^6 + 31*x^7 + 2*x^8)/(1-x^2)^5.
a(n) ~ exp(3*2^(2/3)*Pi*n^(5/6)/5 + 3*Zeta(5)/(4*Pi^4)) / (2^(7/6) * 3^(1/2) * n^(7/12)), where Zeta(5) = A013663. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A255052 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+4,4).

Original entry on oeis.org

1, 5, 30, 145, 660, 2777, 11160, 42805, 158490, 568050, 1980607, 6735380, 22402610, 73022755, 233692345, 735350970, 2278153310, 6956560935, 20958613740, 62354061740, 183332498533, 533074229590, 1533842417185, 4369816273820, 12332669124455, 34495668855729

Offset: 0

Vaclav Kotesovec, Mar 08 2015

nonn

In general, if g.f. = product_{j>=1} 1/(1-x^j)^binomial(j+k-1,k-1), k>=1, then log(a(n)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio

Column k=5 of A075196.

with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*binomial(d+4, 4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # after Alois P. Heinz

nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+4,4],{j,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{j>=1} 1/(1-x^j)^C(j+4,4).

a(n) ~ Pi^(49/288) * exp(25/144 - 105*Zeta(3) / (8*Pi^2) + 5*Zeta'(-3)/12 + 29299*Zeta(5) / (128*Pi^4) + 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 72930375 * Zeta(5)^3 / (2*Pi^14) + 1063324867500 * Zeta(5)^5 / Pi^24 + 41 * 7^(1/6) * Pi * n^(1/6) / (768*3^(1/2)) - 2625 * 3^(1/2) * 7^(1/6) * Zeta(3) * Zeta(5) * n^(1/6) / (2*Pi^7) + 540225 * 3^(1/2) * 7^(1/6) * Zeta(5)^2 * n^(1/6) / (16*Pi^9) - 4740474375 * 3^(1/2) * 7^(1/6) * Zeta(5)^4 * n^(1/6) / (4*Pi^19) + 25 * 7^(1/3) * Zeta(3) * n^(1/3) / (4*Pi^2) - 735 * 7^(1/3) * Zeta(5) * n^(1/3) / (8*Pi^4) + 3969000 * 7^(1/3) * Zeta(5)^3 * n^(1/3) / Pi^14 + 7^(3/2) * Pi * n^(1/2) / (3^(3/2)*8) - 4725 * 21^(1/2) * Zeta(5)^2 * n^(1/2) / Pi^9 + 45 * 7^(2/3) * Zeta(5) * n^(2/3) / (2*Pi^4) + 2 * 3^(1/2) * Pi * n^(5/6) / (5 * 7^(1/6))) / (A^(25/12) * 2^(3/2) * 3^(625/576) * 7^(337/1728) * n^(1201/1728)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 21, 39, 76, 145, 294, 581, 1169, 2276, 4435, 8494, 16237, 30768, 58221, 109466, 205223, 382658, 710808, 1314091, 2420437, 4439753, 8115645, 14781062, 26833241, 48550863, 87575527, 157480827, 282362462, 504819198, 900058558, 1600424247

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n, 3))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

cp's OEIS Frontend

A082544 Number of ordered quintuples (a,b,c,d,e) with gcd(a,b,c,d,e)=1 (1<= {a,b,c,d,e} <= n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A255050 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A267316 Decimal expansion of the Dirichlet eta function at 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A293904 Decimal expansion of zeta(21).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).