A013664 -id:A013664 - cp's OEIS frontend

A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

1, 63, 45991, 2942695, 45982595359, 5109066151, 601081707598999, 38469080386820311, 252396118308232060471, 252395862211967012407, 447134922152359540530757327, 447134770212444455649757327, 2158234586764514215343657417779543, 308319185132349039219686748825649

Offset: 1

Alexander Adamchuk, Jan 16 2008

p divides a(p-1) for prime p > 2. a(n) is prime for n = {19, 47, 164, ...} = A136686.

Lim_{n -> infinity} a(n)/A334605(n) = A275703 = (31/32)*A013664. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = a(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020

Harvey P. Dale, Table of n, a(n) for n = 1..382
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A013664, A058313, A119682, A120296, A136675, A136676.
Cf. A001008, A007406, A007408, A007410, A099828, A103345, A275703.
Cf. A136681, A136682, A136683, A136684, A136685, A136686, A334605 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^6, {k,1,n} ] ], {n,1,30} ]
Accumulate[Table[(-1)^(k+1)/k^6,{k,20}]]//Numerator (* Harvey P. Dale, Aug 21 2023 *)

A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

Original entry on oeis.org

1, 63, 727, 4031, 15559, 45863, 116855, 257983, 526615, 983583, 1755143, 2935231, 4776055, 7407727, 11256623, 16498719, 23859071, 33434063, 46467719, 62949975, 84644439, 111486599, 146142583, 187854119, 240880239, 303814503, 382049919, 473813703, 586746719

Offset: 1

Karl-Heinz Hofmann, May 06 2021

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Joachim von zur Gathen and Jürgen Gerhard, Extract from "3.4. (Non-)Uniqueness of the gcd" chapter, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Column k=6 of A344527.
Cf. A343359, A013664, A018805, A071778, A082540, A082544.
Cf. A342632, A342586, A342935, A342841, A343527, A343193.

a(n)={sum(k=1, n+1, moebius(k)*(n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A343978(n): return sum(mobius(k)*(n//k)**6 for k in range(1, n+1))

from functools import lru_cache
@lru_cache(maxsize=None)
def A343978(n):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A343978(k1)
        j, k1 = j2, n//j2
    return n*(n**5-1)-c+j # Chai Wah Wu, May 17 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^6.
Lim_{n->infinity} a(n)/n^6 = 1/zeta(6) = A343359 = 945/Pi^6.
a(n) = n^6 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

Edited by N. J. A. Sloane, Jun 13 2021

A351268 Sum of the 5th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 33, 244, 33, 3126, 8052, 16808, 33, 244, 103158, 161052, 8052, 371294, 554664, 762744, 33, 1419858, 8052, 2476100, 103158, 4101152, 5314716, 6436344, 8052, 3126, 12252702, 244, 554664, 20511150, 25170552, 28629152, 33, 39296688, 46855314, 52541808, 8052, 69343958

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^5 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 33; a(4) = Sum_{d|4} d^5 * mu(d)^2 = 1^5*1 + 2^5*1 + 4^4*0 = 33.

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

Cf. A008683 (mu), A013661, A013664.

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), this sequence (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^5); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^5 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^5. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^5 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(6)/(6*zeta(2)) = Pi^4/945 = 0.103078... . - Amiram Eldar, Nov 10 2022

A351602 a(n) = n^4 * Sum_{d^2|n} 1 / d^4.

Original entry on oeis.org

1, 16, 81, 272, 625, 1296, 2401, 4352, 6642, 10000, 14641, 22032, 28561, 38416, 50625, 69888, 83521, 106272, 130321, 170000, 194481, 234256, 279841, 352512, 391250, 456976, 538002, 653072, 707281, 810000, 923521, 1118208, 1185921, 1336336, 1500625, 1806624, 1874161, 2085136

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), this sequence (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013664.

f[p_, e_] := p^4*(p^(4*e) - p^(4*Floor[(e - 1)/2]))/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^4*sumdiv(n, d, if (issquare(d), 1/d^2)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^4*(p^(4*e) - p^(4*floor((e-1)/2)))/(p^4 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(6)/5 = Pi^6/4725 = 0.203468... . - Amiram Eldar, Nov 13 2022

A352033 Sum of the 5th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 244, 1, 1, 244, 3126, 1, 244, 1, 16808, 3369, 1, 1, 59293, 1, 3126, 17051, 161052, 1, 244, 3126, 371294, 59293, 16808, 1, 762744, 1, 1, 161295, 1419858, 19933, 59293, 1, 2476100, 371537, 3126, 1, 4101152, 1, 161052, 821793, 6436344, 1, 244, 16808, 9768751

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 3126; a(10) = Sum_{d|10, d<10, d odd} d^5 = 1^5 + 5^5 = 3126.

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), this sequence (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013664, A051002.

Table[Total[Select[Most[Divisors[n]],OddQ]^5],{n,50}] (* Harvey P. Dale, May 01 2023 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e+5) - 1)/(p^5 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^5, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

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

G.f.: Sum_{k>=1} (2*k-1)^5 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051002(n) - n^5*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)-1)/12 = 0.0014452551... . (End)

A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 32, 243, 1024, 3125, 7808, 16807, 32768, 59292, 100032, 161051, 249856, 371293, 537856, 762743, 1048576, 1419857, 1897376, 2476099, 3201024, 4101151, 5153664, 6436343, 7995392, 9768750, 11881408, 14408199, 17211392, 20511149, 24407808, 28629151, 33554432, 39296687

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^5 * Sum_{d|10, d<10, d odd} 1 / d^5 = 10^5 * (1/1^5 + 1/5^5) = 100032.

Robert Israel, 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), this sequence (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013664, A051002.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^5, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..40]); # Robert Israel, Apr 03 2023

A352051[n_]:=DivisorSum[n,1/#^5&,#A352051,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-5, n/2^IntegerExponent[n, 2]] * n^5 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^5 * sigma(n >> valuation(n, 2), -5) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^5 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^5 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051002(n) * A006519(n)^5 - A000035(n).

Sum_{k=1..n} a(k) = c * n^6 / 6, where c = 63*zeta(6)/64 = 1.00144707... . (End)

A068468 Decimal expansion of zeta(6)/(zeta(2)*zeta(3)).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Mar 10 2002

			0.514510101508393123073281186772789616506565746907128.....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for zeta function.

Cf. A013661 (zeta(2)), A002117 (zeta(3)), A013664 (zeta(6)), A082695 (inverse).

R:=RealField(150); SetDefaultRealField(R); L:=RiemannZeta(); 2*Pi(R)^4/(315*Evaluate(L,3)); // G. C. Greubel, Mar 11 2018

RealDigits[Zeta[6]/(Zeta[2]*Zeta[3]), 10, 100][[1]] (* G. C. Greubel, Mar 11 2018 *)

default(realprecision, 100); zeta(6)/(zeta(2)*zeta(3)) \\ G. C. Greubel, Mar 11 2018

From Amiram Eldar, Nov 07 2022: (Start)
Equals 2*Pi^4/(315*zeta(3)).
Equals Product_{p prime} (1 - 1/(p^2-p+1)). (End)

A096960 a(n) = Sum_{0

Original entry on oeis.org

1, 32, 244, 1024, 3126, 7808, 16808, 32768, 59293, 100032, 161052, 249856, 371294, 537856, 762744, 1048576, 1419858, 1897376, 2476100, 3201024, 4101152, 5153664, 6436344, 7995392, 9768751, 11881408, 14408200, 17211392, 20511150

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 32*q^2 + 244*q^3 + 1024*q^4 + 3126*q^5 + 7808*q^6 + 16808*q^7 + 32768*q^8 + ...

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, A013664, A096961, A096962, A096963.

Basis( ModularForms( Gamma0(2), 6), 30) [2]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^5 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^5))};

ModularForms( Gamma0(2), 6, prec=33).gen(1).coefficients(30) # Michael Somos, Jun 04 2013

G.f.: Sum {k>0} k^5 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(5*e) and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 21*zeta(6)/128 = 0.166907... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-5)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

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 *)

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 29, 235, 925, 3101, 6815, 16759, 29597, 57097, 89929, 160931, 217375, 371125, 486011, 728735, 947101, 1419569, 1655813, 2475739, 2868425, 3938365, 4666999, 6435815, 6955295, 9690601, 10762625, 13874563, 15502075, 20510309, 21133315, 28628191, 30307229, 37818785

Offset: 1

Seiichi Manyama, May 18 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A068963, A084218, A372963.
Cf. A001160, A008683.
Cf. A013662, A013664, A088246.
Cf. A350156, A372964.
Cf. A371492, A372952.

f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1); 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)^2*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

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A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

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A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

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A351268 Sum of the 5th powers of the squarefree divisors of n.

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A351602 a(n) = n^4 * Sum_{d^2|n} 1 / d^4.

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A352033 Sum of the 5th powers of the odd proper divisors of n.

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A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

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A068468 Decimal expansion of zeta(6)/(zeta(2)*zeta(3)).

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A096960 a(n) = Sum_{0