A013670 -id:A013670 - cp's OEIS frontend

A161010 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.

1, 8191, 797161, 33550336, 305175781, 6529545751, 16148168401, 137422176256, 423644039001, 2499694822171, 3452271214393, 26745019396096, 25239592216021, 132269647372591, 243274230757741, 562881233944576, 619036127056621

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^13 such that the quotient group Z^13 / 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 13 of A263950.

Cf. A000010, A013670, A000203, A013671, A160897.

f:= proc(n) local t; mul(t[1]^(12*t[2]-12)*(t[1]^13-1)/(t[1]-1), t = ifactors(n)[2]) end proc:
seq(f(n),n=1..100); # Robert Israel, Dec 08 2015

b = 14; Table[Sum[MoebiusMu[n/d] d^(b - 1), {d, Divisors@ n}]/EulerPhi@ n, {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(12*e - 12) * (p^13-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^12, d, if(ispower(d, 13), moebius(sqrtnint(d, 13))*sigma(n^12/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^13 - 1)*f[i,1]^(12*f[i,2] - 12)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_13(n)/J_1(n) where J_13 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^(12e-12) * (p^13-1) / (p-1).
For squarefree n, a(n) = A000203(n^12). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^13, where c = (1/13) * Product_{p prime} (1 + (p^12-1)/((p-1)*p^13)) = 0.14949521105... .
Sum_{k>=1} 1/a(k) = zeta(12)*zeta(13) * Product_{p prime} (1 - 2/p^13 + 1/p^25) = 1.0001233729754... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^13). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A347331 Decimal expansion of 691 * Pi^6 / 675675.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.9831944836800760217380865587210155031...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A008836, A182448, A347328, A347329, A347330.

Cf. A013664, A013670.

RealDigits[691 * Pi^6 / 675675, 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals zeta(12) / zeta(6).
Equals Sum_{k>=1} A008836(k) / k^6.
Equals Product_{p prime} 1/(1+p^(-6)). [corrected by Amiram Eldar, Jun 06 2023]

Data corrected by Amiram Eldar, Jun 06 2023

A017685 Numerator of sum of -11th powers of divisors of n.

Original entry on oeis.org

1, 2049, 177148, 4196353, 48828126, 30248021, 1977326744, 8594130945, 31381236757, 50024415087, 285311670612, 185843885311, 1792160394038, 506442812307, 2883268288216, 17600780175361, 34271896307634, 21433384705031, 116490258898220, 102450026512239, 350279478046112

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017686 (denominator), A013669, A013670.

[Numerator(DivisorSigma(11,n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[Total[Divisors[n]^-11]],{n,20}] (* Harvey P. Dale, Aug 26 2012 *)
Table[Numerator[DivisorSigma[11, n]/n^11], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017686(n) = zeta(11) (A013669).
Dirichlet g.f. of a(n)/A017686(n): zeta(s)*zeta(s+11).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017686(k) = zeta(12) (A013670). (End)

A017687 Numerator of sum of -12th powers of divisors of n.

Original entry on oeis.org

1, 4097, 531442, 16781313, 244140626, 1088658937, 13841287202, 68736258049, 282430067923, 500122072361, 3138428376722, 1486382423891, 23298085122482, 28353876833297, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162, 2048500130460969

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017688 (denominator), A013670, A013671.

[Numerator(DivisorSigma(12,n)/n^12): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[DivisorSigma[12, n]/n^12], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 12)/n^12)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017688(n) = zeta(12) (A013670).
Dirichlet g.f. of a(n)/A017688(n): zeta(s)*zeta(s+12).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017688(k) = zeta(13) (A013671). (End)

A256920 Decimal expansion of Sum_{k>=1} (-1)^k*(zeta(4k)-1) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			-0.07847757966713683831802219324571923504667221732729...
= 1 - 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. Adamchi, H. M. Srivastava, Some series of the zeta and related functions, Analysis (Munich) 18 (1998) 271-288, eq (2.26)
Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A013662, A013666, A013670, A256919.

Join[{0}, RealDigits[1 + (Pi/(2 Sqrt[2]))*(Sin[Pi*Sqrt[2]] + Sinh[Pi*Sqrt[2]]) / (Cos[Pi*Sqrt[2]] - Cosh[Pi*Sqrt[2]]), 10, 105] // First]

1 + (Pi/(2*Sqrt(2)))*(sin(Pi*sqrt(2)) + sinh(Pi*sqrt(2))) / (cos(Pi*sqrt(2)) - cosh(Pi*sqrt(2))).

Equals Sum_{k>=2} 1/(k^4 + 1). - Amiram Eldar, Jul 11 2020

A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 4098, 531444, 16785412, 244140630, 2177857512, 13841287208, 68753047560, 282431130813, 1000488301740, 3138428376732, 8920506494928, 23298085122494, 56721594978384, 129747072969720, 281612482805776, 582622237229778, 1157402774071674

Offset: 0

Seiichi Manyama, Feb 18 2017

Multiplicative because A013959 is. - Andrew Howroyd, Jul 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), this sequence (phi_{12, 1}).
Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).
Cf. A013670.

Table[n * DivisorSigma[11, n], {n, 0, 18}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013959(n) for n > 0.
a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.
Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)

A321815 Sum of 11th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 177148, 1, 48828126, 177148, 1977326744, 1, 31381236757, 48828126, 285311670612, 177148, 1792160394038, 1977326744, 8649804864648, 1, 34271896307634, 31381236757, 116490258898220, 48828126, 350279478046112, 285311670612, 952809757913928

Offset: 1

N. J. A. Sloane, Nov 24 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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=11 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013670, A013959.

List(List(List([1..25],j->DivisorsInt(j)),i->Filtered(i,k->IsOddInt(k))),m->Sum(m,n->n^11)); # Muniru A Asiru, Dec 07 2018

a[n_] := DivisorSum[n, #^11&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321815(n)=sigma(n>>valuation(n,2),11), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321815(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),11)) # Chai Wah Wu, Jul 16 2022

a(n) = A013959(A000265(n)) = sigma_11(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^11*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 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 = zeta(12)/24 = 691*Pi^12/15324309000 = 0.0416769... . (End)

A378768 Squares of powerful numbers that are not prime powers.

Original entry on oeis.org

1296, 5184, 10000, 11664, 20736, 38416, 40000, 46656, 50625, 82944, 104976, 153664, 160000, 186624, 194481, 234256, 250000, 331776, 419904, 455625, 456976, 614656, 640000, 746496, 810000, 937024, 944784, 1000000, 1185921, 1265625, 1327104, 1336336, 1500625, 1679616

Offset: 1

Michael De Vlieger, Dec 06 2024

Contained in A286708, which is a proper subset of A126706.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000290, A001694, A126706, A286708, A372695.

Cf. A013662, A013664, A013670.

With[{nn = 2000}, Select[Rest@ Union[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}] ], Not@*PrimePowerQ]^2]

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A378768(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n)**2 # Chai Wah Wu, Dec 08 2024

a(n) = A286708(n)^2.
Intersection of A000290 and A286708.
Intersection of A000290 and A372695.
Sum_{n>=1} 1/a(n) = zeta(4)*zeta(6)/zeta(12) - Sum_{p prime} (1/(p^4-p^2)) - 1 = 0.0013772572536044025109... . - Amiram Eldar, Dec 10 2024

A369634 Decimal expansion of the infinite product of the Zeta Functions with arguments that are multiples of 3.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 28 2024

Dirichlet generating function of A000688 evaluated at s=3.

			1.22570470512849740952045767158897448248993384223224...

Cf. A000688, A021002, A080729.

Cf. A002117, A013664, A013667, A013670.

evalf(product(Zeta(3*k), k = 1 .. infinity), 120) # Amiram Eldar, Jan 28 2024

prodinf(k=1,zeta(3*k)) \\ Amiram Eldar, Jan 28 2024

Equals Product_{k>=1} zeta(3*k) = A002117 * A013664 * A013667 * A013670 *...

A013688 Continued fraction for zeta(12).

Original entry on oeis.org

1, 4063, 1, 1, 1, 1, 3, 14, 4, 5, 1, 8, 3, 1, 142, 1, 2, 1, 2, 2, 24, 1, 3, 20, 1, 1, 1, 60, 4, 1, 12, 1, 1, 1, 4, 23, 12, 1, 3, 11, 1, 2, 1, 13, 3, 16, 1, 91, 2, 2, 8, 1, 1, 1, 62, 1, 7, 1, 2, 15, 2, 5, 4, 1, 8, 1, 1, 20, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1

Offset: 0

N. J. A. Sloane

Cf. A013670 (decimal expansion).

Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

ContinuedFraction[Zeta[12], 100] (* Paolo Xausa, Jul 03 2024 *)

Offset changed by Andrew Howroyd, Jul 09 2024

cp's OEIS Frontend

A161010 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A347331 Decimal expansion of 691 * Pi^6 / 675675.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A017685 Numerator of sum of -11th powers of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A017687 Numerator of sum of -12th powers of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A256920 Decimal expansion of Sum_{k>=1} (-1)^k*(zeta(4k)-1) (negated).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321815 Sum of 11th powers of odd divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP