A244115 -id:A244115 - cp's OEIS frontend

A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2).

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

Successive derivatives of the zeta function evaluated at x=2 round to (-1)^n * n!, for the n-th derivative, and converge with increasing n. For example, in Mathematica, Derivative[5][Zeta][2] = -120.000824333. A direct formula for the n-th derivative of Zeta at x=2 is: (-1)^n*Sum_{k>=1} log(k)^n/k^2. See also A201994 and A201995. The values of successive derivatives of Zeta(x) as x->1 are given by A252898, and are also related to the factorials. - Richard R. Forberg, Dec 30 2014

			Zeta'(2) = -0.93754825431584375370257409456786497789786028861482...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.18, p. 157.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Huylebrouck, Generalizing Wallis' formula, American Mathematical Monthly, to appear, 2015.
Simon Plouffe, Zeta(1,2) the derivative of Zeta function at 2.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A201994 (2nd derivative), A201995 (3rd derivative), A252898.

Cf. A244115, A261506.

Zeta(1,2); evalf(%, 120); # R. J. Mathar, Oct 10 2011

(* first do *) Needs["NumericalMath`NLimit`"], (* then *) RealDigits[ N[ ND[ Zeta[z], z, 2, WorkingPrecision -> 200, Scale -> 10^-20, Terms -> 20], 111]][[1]] (* Eric W. Weisstein, May 20 2004 *)
(* from version 6 on *) RealDigits[-Zeta'[2], 10, 105] // First (* or *) RealDigits[-Pi^2/6*(EulerGamma - 12*Log[Glaisher] + Log[2*Pi]), 10, 105] // First (* Jean-François Alcover, Apr 11 2013 *)

-zeta'(2) \\ Charles R Greathouse IV, Mar 28 2012

Sum_{n >= 1} log(n) / n^2. - N. J. A. Sloane, Feb 19 2011

Pi^2(gamma + log(2Pi) - 12 log(A))/6, where A is the Glaisher-Kinkelin constant. - Charles R Greathouse IV, May 06 2013

Definition corrected by N. J. A. Sloane, Feb 19 2011

A240966 Decimal expansion of zeta'(-2) (the derivative of Riemann's zeta function at -2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 05 2014

			-0.030448457058393270780251530471154776647000483544973936252971889859...

Eric Weisstein's MathWorld, Riemann Zeta Function.

Cf. A084448 (zeta'(-1)), A075700 (zeta'(0)), A073002 (zeta'(2)), A244115 (zeta'(3)).

Join[{0}, RealDigits[-Zeta[3]/(4*Pi^2), 10, 103] // First]

zeta'(-2) = -zeta(3)/(4*Pi^2).

Equals -log(A243262). - Vaclav Kotesovec, Feb 22 2015

A261506 Decimal expansion of -zeta'(4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2015

			0.06891126589612537984882936558744082715001637487137...

J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Cf. A075700 (0), A073002 (2), A244115 (3).

Cf. A084448 (-1), A240966 (-2), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8).

Flatten[{0, RealDigits[-Zeta'[4], 10, 105][[1]]}]

Sum_{n>=1} log(n) / n^4.

A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965

Offset: 1

Seiichi Manyama, May 21 2024

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Row sums of A350900.
Cf. A057660, A350156.
Cf. A001620, A002117, A013661, A073002, A244115, A306016.

f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)

a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ Amiram Eldar, May 27 2024

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 11 2021

First derivative of the Dirichlet eta function at 3.

			0.0597059061601953583634292662879256783169268731565...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A002117, A091812, A197070, A210593, A244115, A256358.

Flatten[{0, RealDigits[(Log[2] Zeta[3] + 3 Zeta'[3])/4, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^3) \\ Michel Marcus, Nov 11 2021

Equals (log(2) * zeta(3) + 3 * zeta'(3)) / 4.

A365404 The sum of the unitary divisors of the square root of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Sep 03 2023

The number of these divisors is A323308(n).

The sum of the unitary divisors of the largest square dividing n is A365403(n).

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

Cf. A005117, A000188, A034448, A323308, A365403.

Cf. A001620, A002117, A244115.

f[p_, e_] := If[e == 1, 1, p^Floor[e/2] + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^(f[i,2]\2)));}

a(n) = A034448(A000188(n)).
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p) = 1 and a(p^e) = p^floor(e/2) + 1 for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-1).
Sum_{k=1..n} a(k) ~ (n/(2*zeta(3))) * (log(n) + 3*gamma - 1 - 4*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).

A369309 The number of powerful divisors d of n such that n/d is also powerful.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 19 2024

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

Cf. A001694, A005361, A369310.

Cf. A001620, A002117, A078434, A244115.

f[p_,e_] := If[e == 2, 2, e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x==2, 2, x-1), factor(n)[,2]));

Multiplicative with a(p^2) = 2 and a(p^e) = e-1 if e != 2.
a(n) > 0 if and only if n is powerful (A001694).
Dirichlet g.f.: (zeta(2*s)*zeta(3*s)/zeta(6*s))^2.
Sum_{k=1..n} a(k) ~ (zeta(3/2)^2/(2*zeta(3)^2)) * sqrt(n) * (log(n) + 4*gamma - 2 + 6*zeta'(3/2)/zeta(3/2) - 12*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).

A153517 Floor of reciprocal of Zeta'(n), where Zeta'(n) is the derivative of Riemann zeta function.

Original entry on oeis.org

-2, -6, -15, -35, -78, -166, -345, -707, -1435, -2899, -5835, -11721, -23507, -47101, -94318, -188791, -377786, -755845, -1512052, -3024587, -6049818, -12100492, -24202125, -48405772, -96813572, -193629847, -387263296

Offset: 2

Vladimir Reshetnikov, Dec 28 2008

sign

			Floor(1/Zeta'(2)) = -2.

G. C. Greubel, Table of n, a(n) for n = 2..1000

a(2) = floor(1/-A073002), a(3) = floor(1/-A244115), a(4) = floor(1/-A261506).

Mathematica
```
Table[Floor[1/Zeta'[k]], {k, 2, 40}]
```

a(n) = floor(1/zeta'(n)) \\ Iain Fox, Nov 08 2017

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

A309153 a(n) = A000203(n)*A001227(n).

Original entry on oeis.org

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384

Offset: 1

Omar E. Pol, Jul 14 2019

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.

a(n) = A000203(n) iff n is a power of 2.

Amiram Eldar, 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).
Index entries for sequences mentioned by Glaisher.

Cf. A000079, A000203, A001227, A062354, A062355, A064840.

Cf. A001620, A002117, A244115, A306016

Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

cp's OEIS Frontend

A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2).

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A240966 Decimal expansion of zeta'(-2) (the derivative of Riemann's zeta function at -2).

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A261506 Decimal expansion of -zeta'(4).

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A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

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A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

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A365404 The sum of the unitary divisors of the square root of the largest square dividing n.

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A369309 The number of powerful divisors d of n such that n/d is also powerful.

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