A013663 -id:A013663 - cp's OEIS frontend

A059377 Jordan function J_4(n).

1, 15, 80, 240, 624, 1200, 2400, 3840, 6480, 9360, 14640, 19200, 28560, 36000, 49920, 61440, 83520, 97200, 130320, 149760, 192000, 219600, 279840, 307200, 390000, 428400, 524880, 576000, 707280, 748800, 923520, 983040, 1171200, 1252800, 1497600, 1555200, 1874160

Offset: 1

N. J. A. Sloane, Jan 28 2001

This sequence is multiplicative. - Mitch Harris, Apr 19 2005

For n = 4 or n >= 6, a(n) is divisible by 240. - Jianing Song, Apr 06 2019

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

T. D. Noe, Table of n, a(n) for n=1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Michael Lugo, A little number theory problem (2008)
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Wikipedia, Jordan's totient function.

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059378 (J_5), A069091 - A069095 (J_6 through J_10).

Cf. A013663.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end:
seq(J(n,4), n=1..40);

JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 4]; Array[f, 38]
f[p_, e_] := p^(4*e) - p^(4*(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^4*moebius(n/d)),","))

a(n)=if(n<1,0,sumdiv(n,d,d^4*moebius(n/d)))

a(n)=if(n<1,0,dirdiv(vector(n,k,k^4),vector(n,k,1))[n])

{ for (n = 1, 1000, write("b059377.txt", n, " ", sumdiv(n, d, d^4*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009

a(n) = Sum_{d|n} d^4*mu(n/d). - Benoit Cloitre, Apr 05 2002
Multiplicative with a(p^e) = p^(4e)-p^(4(e-1)).
Dirichlet generating function: zeta(s-4)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = Sum_{k=1..n} gcd(k,n)^4 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(n) = n^4*Product_{distinct primes p dividing n} (1 - 1/p^4). - Tom Edgar, Jan 09 2015
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ n^5 / (5*zeta(5)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^4 = 1/zeta(5).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/(p^4-1)^2) = 1.0870036174... (End)
O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 11*x^n + 11*x^(2*n) + x^(3*n))/(1 - x^n)^5 = x + 15*x^2 + 80*x^3 + 240*x^4 + 624*x^5 + .... - Peter Bala, Jan 31 2022
From Peter Bala, Jan 01 2024: (Start)
a(n) = Sum_{d divides n} d * J_3(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_2(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_1(d) * J_3(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n) and J(3,n) = A059376(n).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_3(gcd(k, n)) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_2(gcd(j, k, n)) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^3 * J_1(gcd(i, j, k, n)). (End)
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_2(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_3(j) (apply Lehmer, Theorem 1). - Peter Bala, Jan 29 2024

A085965 Decimal expansion of the prime zeta function at 5.

Original entry on oeis.org

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

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 05 2017

			0.0357550174839242571328...

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..1702
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), this sequence (at 5), A085966 (at 6) to A085969 (at 9).

Cf. A013663, A050997, A242304.

R := RealField(106);
PrimeZeta := func;
[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5,69)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*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, from 1st formula *)
RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,5) \\ Hugo Pfoertner, Feb 03 2020

P(5) = Sum_{p prime} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.
Equals 1/2^5 + A085994 + A086035. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A050997(k). - Amiram Eldar, Jul 27 2020

A013671 Decimal expansion of zeta(13).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.0001227133475784891467518365263573957142751058955098451367026716208967...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A013663, A013667, A013669, A013671, A013675, A013677.

RealDigits[Zeta[13],10,120][[1]] (* Harvey P. Dale, Dec 24 2016 *)

zeta(13) \\ Charles R Greathouse IV, Apr 25 2016

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(13) = sum {n >= 1} 1/n^13.
zeta(13) = 2^13/(2^13 - 1)*( sum {n even} n^9*p(n)*p(1/n)/(n^2 - 1)^14 ), where p(n) = n^6 + 21*n^4 + 35*n^2 + 7. (End)
zeta(13) = Sum_{n >= 1} (A010052(n)/n^(13/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(13/2) ). - Mikael Aaltonen, Feb 22 2015
zeta(13) = Product_{k>=1} 1/(1 - 1/prime(k)^13). - Vaclav Kotesovec, May 02 2020

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Original entry on oeis.org

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (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. 22.

Cf. A023872, A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, 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^3),{k,1,nmax}],{x,0,nmax}],x]

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018
Euler transform of A309335. - Georg Fischer, Nov 10 2020

A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

Original entry on oeis.org

1, 1, 9, 36, 136, 477, 1703, 5746, 19099, 61622, 195366, 607069, 1856516, 5586870, 16579850, 48549116, 140438966, 401592524, 1136121837, 3181700219, 8825733603, 24261363403, 66124058839, 178757752892, 479513547399, 1276792213203, 3375707760306, 8864712158225

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5422 (first 1001 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=3 of A144048.

Cf. A248882.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^3: 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^3, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

max = 27; Series[ Product[ 1/(1-x^k)^k^3, {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^3)) \\ G. C. Greubel, Oct 30 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^3)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

a(n) ~ (3*Zeta(5))^(59/600) * exp(5 * n^(4/5) * (3*Zeta(5))^(1/5) / 2^(7/5) + Zeta'(-3)) / (2^(41/200) * n^(359/600) * sqrt(5*Pi)), where Zeta(5) = A013663 = 1.036927755143369926..., Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.00537857635777430114441697421... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_4(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k). - Seiichi Manyama, Mar 04 2017

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

A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, 1, 4, 14, 36, 101, 260, 669, 1669, 4116, 9932, 23636, 55483, 128532, 294422, 667026, 1496232, 3324720, 7323570, 15998749, 34679966, 74622839, 159454379, 338472749, 713956569, 1496950669, 3120663129, 6469901522, 13343153563, 27379250529, 55907749171

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A248882, A028377, A258341, A258342, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018

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

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 28 2018

A013675 Decimal expansion of zeta(17).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.0000076371976378997622736002935630292130882490902626790953798439729356...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 811.

Cf. A013663, A013667, A013669, A013671, A013675, A013677, A010057.

RealDigits[Zeta[17], 10, 75][[1]] (* Vincenzo Librandi, Feb 24 2015 *)

zeta(17) \\ Charles R Greathouse IV, Dec 04 2013

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(17) = sum {n >= 1} 1/n^17.
zeta(17) = 2^17/(2^17 - 1)*( sum {n even} n^11*p(n)*p(1/n)/(n^2 - 1)^18 ), where p(n) = n^8 + 36*n^6 + 126*n^4 + 84*n^2 + 9. Cf. A013663, A013667 and A013671.
(End)
zeta(17) = Sum_{n >= 1} (A010052(n)/n^(17/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(17/2) ). - Mikael Aaltonen, Feb 23 2015
zeta(17) = Product_{k>=1} 1/(1 - 1/prime(k)^17). - Vaclav Kotesovec, May 02 2020

A013677 Decimal expansion of zeta(19).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.0000019082127165539389256569577951013532585711448386302359330467618239...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A013663, A013667, A013669, A013671, A013675.

RealDigits[Zeta[19], 10, 75][[1]] (* Vincenzo Librandi, Feb 24 2015 *)

zeta(19) = Sum_{n >= 1} (A010052(n)/n^(19/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(19/2) ). - Mikael Aaltonen, Feb 23 2015

zeta(19) = Product_{k>=1} 1/(1 - 1/prime(k)^19). - Vaclav Kotesovec, May 02 2020

A152651 Decimal expansion of 3Zeta(5) - Zeta(3)Pi^2/6.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 10 2008

A division by 2 is missing in Mezo's penultimate formula on page 4.

			Equals 1.1334789151328136607970110178859769320890912918456042272...

David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995) 1191-1198.
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.

RealDigits[3*Zeta[5]-Zeta[3]*Pi^2/6,10,120][[1]] (* Harvey P. Dale, Apr 29 2019 *)

3*zeta(5) - zeta(3)*Pi^2/6 \\ Michel Marcus, Jul 07 2015

Equals Sum_(j >= 1) H(j)/j^4 = where H(j) = A001008(j)/A002805(j).

Equals 3*A013663 - A002117*A013661.

A255323 Product_{k=1..n} k^(k^4).

Original entry on oeis.org

1, 65536, 29060398333495723291328487792256607374737408

Offset: 1

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(4) has 198 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 1..5

Cf. A002109, A051675, A255321, A255344, A013663, A243264.

Table[Product[k^(k^4), {k, 1, n}], {n, 1, 5}]

a(n)=prod(k=1,n,k^k^4) \\ Charles R Greathouse IV, Sep 08 2015

a(n) ~ A243264 * n^(n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30) / exp(n^5/25 - n^3/12 + 13*n/360), where A243264 = exp(-3*Zeta(5)/(4*Pi^4)).

cp's OEIS Frontend

A059377 Jordan function J_4(n).

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A085965 Decimal expansion of the prime zeta function at 5.

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A013671 Decimal expansion of zeta(13).

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A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

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A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).

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Formula

A013675 Decimal expansion of zeta(17).

A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

A152651 Decimal expansion of 3Zeta(5) - Zeta(3)Pi^2/6.