A013668 -id:A013668 - cp's OEIS frontend

A266556 Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).

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

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 9th Bendersky constant.

			1.018469929920992912170659049376672172308610190564074920380...

G. C. Greubel, Table of n, a(n) for n = 1..2002
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)), A266554 (A(7)), 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, A013668, A266260, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[10]/10)*(EulerGamma + Log[2*Pi] - Zeta'[10]/Zeta[10]), 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(9) = exp(H(9)*B(10)/10 - zeta'(-9)) = exp((B(10)/10)*(EulerGamma + log(2*Pi) - (zeta'(10)/zeta(10)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^10-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(10)/10 = 1/132 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266557 Decimal expansion of the generalized Glaisher-Kinkelin constant A(10).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 10th Bendersky constant.

			1.01911023332938385372216470498629751351348137284099604...

G. C. Greubel, Table of n, a(n) for n = 1..2002

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)), A266555 (A(8)), A266556 (A(9)), 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. A013668, A013669, A266261, A027641, A027642.

Exp[N[(BernoulliB[10]/4)*(Zeta[11]/Zeta[10]), 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(10) = exp(-zeta'(-10)) = exp((B(10)/4)*(zeta(11)/zeta(10))).
A(10) = exp(10! * Zeta(11) / (2^11 * Pi^10)). - Vaclav Kotesovec, Jan 01 2016

A351272 Sum of the 9th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 513, 19684, 513, 1953126, 10097892, 40353608, 513, 19684, 1001953638, 2357947692, 10097892, 10604499374, 20701400904, 38445332184, 513, 118587876498, 10097892, 322687697780, 1001953638, 794320419872, 1209627165996, 1801152661464, 10097892, 1953126, 5440108178862

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^9 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 513; a(4) = Sum_{d|4} d^9 * mu(d)^2 = 1^9*1 + 2^9*1 + 4^9*0 = 513.

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

Cf. A008683 (mu), A013661, A013668.

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), A351270 (k=7), A351271 (k=8), this sequence (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^9); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^9],{n,30}] (* Harvey P. Dale, Feb 21 2023 *)

a(n) = Sum_{d|n} d^9 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^9. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^9 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/(10*zeta(2)) = Pi^8/155925 = 0.0608531... . - Amiram Eldar, Nov 10 2022

A351606 a(n) = n^8 * Sum_{d^2|n} 1 / d^8.

Original entry on oeis.org

1, 256, 6561, 65792, 390625, 1679616, 5764801, 16842752, 43053282, 100000000, 214358881, 431661312, 815730721, 1475789056, 2562890625, 4311810048, 6975757441, 11021640192, 16983563041, 25700000000, 37822859361, 54875873536, 78310985281, 110505295872, 152588281250

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

Cf. A013668.

f[p_, e_] := p^8*(p^(8*e) - p^(8*Floor[(e - 1)/2]))/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^8*sumdiv(n, d, if (issquare(d), 1/d^4)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^8*(p^(8*e) - p^(8*floor((e-1)/2)))/(p^8 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(10)/9 = Pi^10/841995 = 0.1112216... . - Amiram Eldar, Nov 13 2022

A352037 Sum of the 9th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 19684, 1, 1, 19684, 1953126, 1, 19684, 1, 40353608, 1972809, 1, 1, 387440173, 1, 1953126, 40373291, 2357947692, 1, 19684, 1953126, 10604499374, 387440173, 40353608, 1, 38445332184, 1, 1, 2357967375, 118587876498, 42306733, 387440173, 1, 322687697780

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 1953126; a(10) = Sum_{d|10, d<10, d odd} d^9 = 1^9 + 5^9 = 1953126.

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), A352035 (k=7), A352036 (k=8), this sequence (k=9), A352038 (k=10).

Cf. A000035, A013668, A321813.

f[2, e_] := 1; f[p_, e_] := (p^(9*e+9) - 1)/(p^9 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^9, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

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

G.f.: Sum_{k>=1} (2*k-1)^9 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321813(n) - n^9*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)-1)/20 = 0.0000497287... . (End)

A352055 Sum of the 9th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 512, 19683, 262144, 1953125, 10078208, 40353607, 134217728, 387440172, 1000000512, 2357947691, 5160042496, 10604499373, 20661047296, 38445332183, 68719476736, 118587876497, 198369368576, 322687697779, 512000262144, 794320419871, 1207269218304, 1801152661463, 2641941757952

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^9 * Sum_{d|10, d<10, d odd} 1 / d^9 = 10^9 * (1/1^9 + 1/5^9) = 1000000512.

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), A352053 (k=7), A352054 (k=8), this sequence (k=9), A352056 (k=10).

Cf. A000035, A006519, A013668, A321813.

A352055[n_]:=DivisorSum[n,1/#^9&,#A352055,50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-9, n/2^IntegerExponent[n, 2]] * n^9 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^9 * sigma(n >> valuation(n, 2), -9) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^9 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^9 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321813(n) * A006519(n)^9 - A000035(n).

Sum_{k=1..n} a(k) = c * n^10 / 10, where c = 1023*zeta(10)/1024 = 1.0000170413... . (End)

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 *)

A354600 a(n) = Product_{k=0..9} floor((n+k)/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1536, 2304, 3456, 5184, 7776, 11664, 17496, 26244, 39366, 59049, 78732, 104976, 139968, 186624, 248832, 331776, 442368, 589824, 786432, 1048576, 1310720, 1638400, 2048000, 2560000, 3200000, 4000000

Offset: 0

Wesley Ivan Hurt, Jul 08 2022

For n >= 10, a(n) is the maximal product of ten positive integers with sum n.

Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 9, -18, 9, 0, 0, 0, 0, 0, 0, 0, -36, 72, -36, 0, 0, 0, 0, 0, 0, 0, 84, -168, 84, 0, 0, 0, 0, 0, 0, 0, -126, 252, -126, 0, 0, 0, 0, 0, 0, 0, 126, -252, 126, 0, 0, 0, 0, 0, 0, 0, -84, 168, -84, 0, 0, 0, 0, 0, 0, 0, 36, -72, 36, 0, 0, 0, 0, 0, 0, 0, -9, 18, -9, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).
Index entries for subsets of {1,..,n} with disallowed differences

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), A009694 (k=8), A009714 (k=9), this sequence (k=10).

Cf. A008454 (subsequence), A013668.

Table[Product[Floor[(n + k)/10], {k, 0, 9}], {n, 0, 50}]

a(n) = prod(k=0, 9, (n+k)\10); \\ Michel Marcus, Jul 09 2022

a(n) = 2*a(n-1) - a(n-2) + 9*a(n-10) - 18*a(n-11) + 9*a(n-12) - 36*a(n-20) + 72*a(n-21) - 36*a(n-22) + 84*a(n-30) - 168*a(n-31) + 84*a(n-32) - 126*a(n-40) + 252*a(n-41) - 126*a(n-42) + 126*a(n-50) - 252*a(n-51) + 126*a(n-52) - 84*a(n-60) + 168*a(n-61) - 84*a(n-62) + 36*a(n-70) - 72*a(n-71) + 36*a(n-72) - 9*a(n-80) + 18*a(n-81) - 9*a(n-82) + a(n-90) - 2*a(n-91) + a(n-92).
Sum_{n>=10} 1/a(n) = 1 + zeta(10). - Amiram Eldar, Jan 10 2023
a(10*n) = n^10 (A008454). - Bernard Schott, Feb 02 2023

A069094 Jordan function J_9(n).

Original entry on oeis.org

1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A013668.

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
f[p_, e_] := p^(9*e) - p^(9*(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^9*moebius(n/d)),","))

a(n) = Sum_{d|n} d^9*mu(n/d).
Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).
Dirichlet generating function: zeta(s-9)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252...  (End)

A096962 a(n) = Sum_{0

Original entry on oeis.org

1, 512, 19684, 262144, 1953126, 10078208, 40353608, 134217728, 387440173, 1000000512, 2357947692, 5160042496, 10604499374, 20661047296, 38445332184, 68719476736, 118587876498, 198369368576, 322687697780, 512000262144

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 512*q^2 + 19684*q^3 + 262144*q^4 + 1953126*q^5 + 10078208*q^6 + ...

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.

Cf. A007331, A013668, A096960, A096961, A096963.

A := Basis( ModularForms( Gamma0(2), 10), 21); A[2] + 512*A[3]; /* Michael Somos, Aug 25 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^9 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^4, u4 = QPochhammer[ q^4]^8}, q u2 (u1 + 32 q u4) (u1^2 + 496 q u4 u1 + 7936 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^9))}; /* Michael Somos, Jun 04 2013 */

{a(n) = local(A, A1, A2, A4); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^4; A4 = eta(x^4 + A)^8; polcoeff( A2 * (A1 + 32*x * A4) * (A1^2 + 496*x * A1*A4 + 7936*x^2 * A4^2) / A1, n))}; /* Michael Somos, Jun 04 2013 */

ModularForms( Gamma0(2), 10, prec=33).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^9 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(9*e) and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = 1023*zeta(10)/10240 = 31*Pi^10/29030400 = 0.100001704136... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-9)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

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A266556 Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).

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A266557 Decimal expansion of the generalized Glaisher-Kinkelin constant A(10).

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A351272 Sum of the 9th powers of the squarefree divisors of n.

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A351606 a(n) = n^8 * Sum_{d^2|n} 1 / d^8.

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A352037 Sum of the 9th powers of the odd proper divisors of n.

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A352055 Sum of the 9th powers of the divisor complements of the odd proper divisors of n.

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A293904 Decimal expansion of zeta(21).

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A354600 a(n) = Product_{k=0..9} floor((n+k)/10).

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