A261506 -id:A261506 - 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

A244115 Decimal expansion of -zeta'(3) (the first derivative of the zeta function at 3).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jun 20 2014

			0.19812624288563685333068182150328579687554279346383500334688996319272566942265...

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

Cf. A073002, A261506, A288391.

Mathematica
```
RealDigits[-Zeta'[3], 10, 105][[1]]
```

Sum_{n>=1} log(n) / n^3. - Vaclav Kotesovec, Aug 22 2015

A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

Original entry on oeis.org

1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660

Offset: 1

Ilya Gutkovskiy, Aug 29 2019

Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
Dirichlet convolution of A008966 with A018804.
Dirichlet convolution of A038040 with A271102.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Dedekind Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A001615, A008966, A018804, A038040, A271102.

Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.

f:= proc(n) local t;
  mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022

Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
a(p) = 2*p, where p is prime.
Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019
a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019
a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

Original entry on oeis.org

1, 15, 53, 176, 249, 795, 685, 1856, 2133, 3735, 2661, 9328, 4393, 10275, 13197, 18432, 9825, 31995, 13717, 43824, 36305, 39915, 24333, 98368, 46625, 65895, 76545, 120560, 48777, 197955, 59581, 176128, 141033, 147375, 170565, 375408, 101305, 205755, 232829, 462144

Offset: 1

Seiichi Manyama, May 17 2024

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

Column k=3 of A372938.
Cf. A343497, A368743, A372929, A372930.
Cf. A000005, A008683.
Cf. A001620, A013662, A261506.

f[p_, e_] := (e - e/p^3 + 1) * p^(3*e); 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)*d^3*numdiv(d));

a(n) = Sum_{d|n} mu(n/d) * d^3 * tau(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (e - e/p^3 + 1) * p^(3*e).
Dirichlet g.f.: zeta(s-3)^2/zeta(s).
Sum_{k=1..n} a(k) ~ (n^4/(4*zeta(4))) * (log(n) + 2*gamma - 1/4 - zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)

A340440 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 07 2021

			1.0231387264279392955...

R. J. Mathar, The series limit of sum_k 1/(k log k (log log k)^2), arXiv:0902.0789 [math.NA], 2009-2021, version 3, App. B.

Cf. A073002, A261506, A340485, A340484.

sumpos(k=2, log(k)/(k^2-1)) \\ Michel Marcus, Jan 09 2021

Equals Sum_{i>=1} -zeta'(2i) = A073002 + A261506 - Sum_{i>=3} zeta'(2i).
Sum_{k>=2} log(k)/(k^2-s) = -Sum_{i>=1} s^(i-1)*zeta'(2i) for |s|<4. - R. J. Mathar, May 03 2021
Equals log(2)/2 + Sum_{k>=1} (zeta(2*k)-1)/(2*k-1). - Amiram Eldar, Jun 08 2021

A340485 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1)^2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 09 2021

			0.10732537164203023969506024850218288032472798982043615...

Cf. A261506, A340440, A340484.

evalf(-Zeta'(4) - Sum(i * Zeta'(2*i+2), i = 2 .. infinity), 120); # Amiram Eldar, Mar 09 2024

sumpos(k=2, log(k)/(k^2-1)^2) \\ Michel Marcus, Jan 09 2021

-zeta'(4) - sumpos(i=2, i*zeta'(2*i+2)) \\ Amiram Eldar, Mar 09 2024

Equals -Sum_{i>=1} i*zeta'(2*i+2) = A261506 - Sum_{i>=2} i*zeta'(2*i+2).

More terms from Amiram Eldar, Mar 09 2024

A368883 The number of infinitary divisors of n that are cubefree.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 09 2024

The number of infinitary divisors of n that are squarefree is A055076(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000583, A004709, A037445, A055076, A077609.

f[p_, e_] := Switch[Mod[e, 4], 1, 2, 2, 2, 3, 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> [1,2,2,3][x%4+1], factor(n)[, 2]));

Multiplicative with a(p^e) = 2 if e == 1 or 2 (mod 4), 3 if e == 3 (mod 4), and 1 if e == 0 (mod 4).
a(n) >= 1, with equality if and only if n is a 4th power (A000583).
a(n) <= A037445(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 2/p^s + 2/p^(2*s) + 3/p^(3*s)).
From Vaclav Kotesovec, Jan 09 2024: (Start)
Dirichlet g.f.: zeta(4*s) * zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).
Sum_{k=1..n} a(k) ~ f(1) * zeta(4) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1) + 4*zeta'(4)/zeta(4)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 4/p^4 + 3/p^5) = 0.5857770602270641007515615375942370402509903724261557972367075945186871...,
f'(1) = f(1) * Sum_{p prime} (2*p^2 - p + 15) * log(p) / (p^4 + p^3 + p - 3) = f(1) * 1.319786264712492218167871116508220489817987315752197198819256094...,
gamma is the Euler-Mascheroni constant A001620, zeta(4) = Pi^4/90 = A013662 and for zeta'(4) see A261506. (End)

A045771 Number of similar sublattices of index n^2 in root lattice D_4.

Original entry on oeis.org

1, 1, 8, 1, 12, 8, 16, 1, 41, 12, 24, 8, 28, 16, 96, 1, 36, 41, 40, 12, 128, 24, 48, 8, 97, 28, 176, 16, 60, 96, 64, 1, 192, 36, 192, 41, 76, 40, 224, 12, 84, 128, 88, 24, 492, 48, 96, 8, 177, 97, 288, 28, 108, 176, 288, 16, 320, 60, 120, 96, 124, 64, 656, 1

Offset: 1

Michael Baake (baake(AT)miles.math.ualberta.ca)

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math., Vol. 51, No. 6 (1999), 1258-1276.
Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Index entries for sequences related to D_4 lattice.
Index entries for sequences related to sublattices.

Cf. A001620, A013662, A035292, A261506, A306016.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[1 <= p <= 2, 1, (e + 1) p^e + (2 (1 + (e p - e - 1)*p^e))/((p - 1)^2)]] &, 64] (*  Michael De Vlieger, Mar 02 2018 *)

fp(p, e) = if (p % 2, (e+1)*p^e + 2*(1-(e+1)*p^e+e*p^(e+1))/(p-1)^2, 1);
a(n) = { my(f = factor(n)); prod(i=1, #f~, fp(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014

Multiplicative with a(2^p) = 1, a(p^e) = (e+1)*p^e + (2*(1+(e*p-e-1)*p^e))/((p-1)^2), p>2. - Christian G. Bower, May 21 2005
From Amiram Eldar, May 26 2025: (Start)
Dirichlet g.f.: (zeta(s-1)^2 * zeta(s)^2 / zeta(2*s)) * (1 - 1/2^(s-1))^2/(1 + 1/2^s).
Sum_{k=1..n} a(k) ~ (n^2/4)*(log(n) + 2*gamma - 1/2 + 11*log(2)/5 + 2*zeta'(2)/zeta(2) - 2*zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)

More terms from Michel Marcus, Mar 03 2014

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)
```

cp's OEIS Frontend

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

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A244115 Decimal expansion of -zeta'(3) (the first derivative of the zeta function at 3).

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A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

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A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

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A340440 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1).

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A340485 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1)^2.

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A368883 The number of infinitary divisors of n that are cubefree.

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