A013669 -id:A013669 - cp's OEIS frontend

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

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

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

A069095 Jordan function J_10(n).

Original entry on oeis.org

1, 1023, 59048, 1047552, 9765624, 60406104, 282475248, 1072693248, 3486725352, 9990233352, 25937424600, 61855850496, 137858491848, 288972178704, 576640565952, 1098437885952, 2015993900448, 3566920035096, 6131066257800

Offset: 1

Benoit Cloitre, Apr 05 2002

a(n) is divisible by 264 = (2^3)*3*11 = A006863(5), except for n = 1, 2, 3 or 11. See Lugo. - Peter Bala, Jan 13 2024

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

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J-2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069091 - A069094 (J_6 through J_9).

Cf. A013669.

f:= n -> n^10*mul(1-1/p^10, p=numtheory:-factorset(n)):
map(f, [$1..30]); # Robert Israel, Jan 09 2015

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 10]; Array[f, 21]
f[p_, e_] := p^(10*e) - p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI
```
a(n) = sumdiv(n,d,d^10*moebius(n/d));
```

a(n) = Sum_{d|n} d^10*mu(n/d).
Multiplicative with a(p^e) = p^(10e)-p^(10(e-1)).
Dirichlet generating function: zeta(s-10)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^10*Product_{distinct primes p dividing n} (1-1/p^10). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^11 / (11*zeta(11)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^10 = 1/zeta(11).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^10/(p^10-1)^2) = 1.0009955309... (End)

A013958 a(n) = sigma_10(n), the sum of the 10th powers of the divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1049601, 9765626, 60526250, 282475250, 1074791425, 3486843451, 10009766650, 25937424602, 61978939050, 137858491850, 289537131250, 576660215300, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 10250010815226, 16680163512500, 26585860217050

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013669, A013954-A013972, A017665-A017712.

[DivisorSigma(10,n): n in [1..20]]; // Bruno Berselli, Apr 10 2013

lst={};Do[AppendTo[lst,DivisorSigma[10,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[10,Range[20]] (* Harvey P. Dale, Jan 04 2012 *)

a(n)=sigma(n,10) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,10)for n in range(1,18)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^10*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^9)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10)-1)/(p^10-1).
Dirichlet g.f.: zeta(s)*zeta(s-10).
Sum_{k=1..n} a(k) = zeta(11) * n^11 / 11 + O(n^12). (End)

A079395 a(n) = prime(n)^11.

Original entry on oeis.org

2048, 177147, 48828125, 1977326743, 285311670611, 1792160394037, 34271896307633, 116490258898219, 952809757913927, 12200509765705829, 25408476896404831, 177917621779460413, 550329031716248441

Offset: 1

Jon Perry, Jan 06 2003

			2^11 = 2048.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Subsequence of A008455.

Cf. A013669.

[p^11: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A079395:=n->ithprime(n)^11: seq(A079395(n), n=1..30); # Wesley Ivan Hurt, Feb 07 2017

Array[Prime[ # ]^11 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Prime[Range[20]]^11 (* Vincenzo Librandi, Mar 27 2014 *)

PARI
```
forprime (p=2,100,print1(p^11","))
```

From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(11)/zeta(22).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(11) = 1/A013669. (End)

A351273 Sum of the 10th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1025, 9765626, 60526250, 282475250, 1025, 59050, 10009766650, 25937424602, 60526250, 137858491850, 289537131250, 576660215300, 1025, 2015993900450, 60526250, 6131066257802, 10009766650, 16680163512500, 26585860217050, 41426511213650, 60526250

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^10 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 1025; a(4) = Sum_{d|4} d^10 * mu(d)^2 = 1^10*1 + 2^10*1 + 4^10*0 = 1025.

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

Cf. A008683 (mu), A013661, A013669.

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), A351272 (k=9), this sequence (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^10); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^10],{n,25}] (* Harvey P. Dale, Nov 20 2022 *)

a(n) = Sum_{d|n} d^10 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^10. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^10 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/(11*zeta(2)) = 0.0552934... . - Amiram Eldar, Nov 10 2022

A351607 a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

Original entry on oeis.org

1, 512, 19683, 262656, 1953125, 10077696, 40353607, 134479872, 387440172, 1000000000, 2357947691, 5169858048, 10604499373, 20661046784, 38443359375, 68853956608, 118587876497, 198369368064, 322687697779, 513000000000, 794280046581, 1207269217792, 1801152661463

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), A351606 (k=8), this sequence (k=9), A351608 (k=10).

Cf. A013669.

f[p_, e_] := p^9*(p^(9*e) - p^(9*Floor[(e - 1)/2]))/(p^9 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^9*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^9))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^9*(p^(9*e) - p^(9*floor((e-1)/2)))/(p^9 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(11)/10 = 0.100049... . - Amiram Eldar, Nov 13 2022

A352038 Sum of the 10th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 59050, 1, 1, 59050, 9765626, 1, 59050, 1, 282475250, 9824675, 1, 1, 3486843451, 1, 9765626, 282534299, 25937424602, 1, 59050, 9765626, 137858491850, 3486843451, 282475250, 1, 576660215300, 1, 1, 25937483651, 2015993900450, 292240875, 3486843451

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 9765626; a(10) = Sum_{d|10, d<10, d odd} d^10 = 1^10 + 5^10 = 9765626.

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), A352037 (k=9), this sequence (k=10).

Cf. A000035, A013669, A321814.

f[2, e_] := 1; f[p_, e_] := (p^(10*e+10) - 1)/(p^10 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^10, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

from math import prod
from sympy import factorint
def A352038(n): return prod((p**(10*(e+1))-1)//(p**10-1) for p, e in factorint(n).items() if p > 2) - (n**10 if n % 2 else 0) # Chai Wah Wu, Mar 01 2022

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

G.f.: Sum_{k>=1} (2*k-1)^10 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321814(n) - n^10*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^11, where c = (zeta(11)-1)/22 = 0.0000224631... . (End)

A352056 Sum of the 10th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 1024, 59049, 1048576, 9765625, 60467200, 282475249, 1073741824, 3486843450, 10000001024, 25937424601, 61918412800, 137858491849, 289254656000, 576660215299, 1099511627776, 2015993900449, 3570527693824, 6131066257801, 10240001048576, 16680163512499

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^10 * Sum_{d|10, d<10, d odd} 1 / d^10 = 10^10 * (1/1^10 + 1/5^10) = 10000001024.

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), A352055 (k=9), this sequence (k=10).

Cf. A000035, A006519, A013669, A321814.

A352056[n_]:=DivisorSum[n,1/#^10&,#A352056,50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-10, n/2^IntegerExponent[n, 2]] * n^10 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^10 * sigma(n >> valuation(n, 2), -10) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^10 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^10 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321814(n) * A006519(n)^10 - A000035(n).

Sum_{k=1..n} a(k) = c * n^11 / 11, where c = 2047*zeta(11)/2048 = 1.00000566605... . (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 *)

Previous Showing 11-20 of 45 results. Next

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A013677 Decimal expansion of zeta(19).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A069095 Jordan function J_10(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A013958 a(n) = sigma_10(n), the sum of the 10th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A079395 a(n) = prime(n)^11.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A351273 Sum of the 10th powers of the squarefree divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A351607 a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

Views

Author

Keywords

Links

Crossrefs

Programs