A073002 -id:A073002 - cp's OEIS frontend

A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

1, 17, 53, 197, 297, 873, 1069, 2093, 2822, 4422, 4906, 10090, 10766, 13902, 17502, 23902, 25058, 36722, 38166, 52566, 59622, 67366, 69482, 106346, 111971, 122787, 134451, 162675, 166039, 223639, 227483, 264347, 281771, 300267, 319867, 424843, 430319, 453423

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320896.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k^2*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A329117 Decimal expansion of Sum_{k>=1} (k^(1/k^2) - 1).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Nov 05 2019

			0.971499034283308757222625062314754580022...

Cf. A073002, A098572.

digits = 120; d = 1; j = 2; s = Pi^2 * (2*Log[Glaisher] - Log[2*Pi]/6 - EulerGamma/6); While[Abs[d] > 10^(-digits - 5), d = (-1)^j/j!*Derivative[j][Zeta][2*j]; s += d; j++]; RealDigits[s, 10, 120][[1]] (* Vaclav Kotesovec, Jun 18 2023 *)

sumpos(k=1, k^(1/k^2) - 1) \\ Michel Marcus, Nov 05 2019

Equals Sum_{k>=1} (-1)^k / k! * k-th derivative of zeta(2*k). - Vaclav Kotesovec, Jun 18 2023

A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

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

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3 , 2, e + 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,1] <= 3, 2, f[i,2]+1));}

Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000005(A065330(n)) * A034444(A065331(n)).
Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.
Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

A252898 Decimal expansion of lim_{n->infinity} -FractionalPart[Zeta'(1+1/n)] or -FractionalPart[Zeta'(1-1/n)], where Zeta' is the first derivative of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Richard R. Forberg, Dec 24 2014

Zeta'(x) -> negative infinity as x -> 1, from above and below.
When 1 is approached using arguments of (1+1/n) or (1-1/n), its fractional part converges to this constant.
The Euler-Mascheroni constant is the fractional part as x->1 for Zeta(x), but with a different symmetry approaching 1 from above vs. below. See A001620 and below.
The integer part of Zeta'(1 + 1/n) or Zeta'(1 - 1/n)  =  -(n^2 - 1).
Corresponding constants, as taken from the fractional part, exist for the higher order derivatives of the Riemann zeta as x->1 with these arguments. The list below shows converged values up to the 10th derivative approaching 1 from above, using
x = 1 + 1/n, as n -> infinity, with signs:
  Derivative[1] = -0.9271841545163232751394136...  (this entry)
  Derivative[2] =  0.9903096368071276815154696...
  Derivative[3] = -0.0020538344203033458661600...
  Derivative[4] =  0.0023253700654673000057468...
  Derivative[5] = -0.0007933238173010627017533...
  Derivative[6] =  0.9997612306545698003901275...
  Derivative[7] = -0.9994727104329422489539259...
  Derivative[8] =  0.9996478766461969604903979...
  Derivative[9] = -0.9999656052255819119518220...
  Derivative[10]=  0.0002053328149090647946837...
Even order derivatives, D[2m], (e.g., 2nd, 4th, 6th, ...) have different fractional values when approaching 1 from below equal to: -(1-D[2m]). The same is true for D[0], or Zeta itself.
The integer sequences associated with the integer part, with x ->1 from above and starting with the argument x= 2 = 1+1/n, hence n = 1 to infinity, are:
  Derivative[1] =  -(n^2-1)
  Derivative[2] =   (2!*n^3-1)
  Derivative[3] =  -(3!*n^4)
  Derivative[4] =   (4!*n^5)
  Derivative[5] =  -(5!*n^6)
  Derivative[6] =   (6!*n^7-1), except at n=1, where value = 720 with fract ~0.0001
  Derivative[7] =  -(7!*n^8-1)
  Derivative[8] =   (8!*n^9-1)
  Derivative[9] =  -(9!*n^10-1)
  Derivative[10] =  (10!*n^11)
Thus, rounding the m-th derivative of Zeta(x) at x=2 (n=1) gives (-1)^m * m! for m>=1. See A073002.

			0.9271841545163232751394136...

Cf. A001620, A073002, A082633.

s:= convert(evalf(1+gamma(1), 140), string):
seq(parse(s[n+2]), n=0..110);  # Alois P. Heinz, Dec 30 2014

FractionalPart[N[Derivative[1][Zeta][
   1 + 1/(1000000000000000000000000000000000000000000000000000000000000)], 400]]

Limit_{n -> infinity} -FractionalPart[Zeta'(1+1/n)]
Limit_{n -> infinity} -FractionalPart[Zeta'(1-1/n)]
Equals 1 - A082633. - Alois P. Heinz, Dec 30 2014

More digits from Alois P. Heinz, Dec 30 2014

A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 21, 40, 22, 60, 26, 56, 60, 46, 34, 84, 38, 100, 84, 88, 46, 132, 55, 104, 66, 140, 58, 240, 62, 94, 132, 136, 140, 210, 74, 152, 156, 220, 82, 336, 86, 220, 210, 184, 94, 276, 105, 220, 204, 260, 106, 264, 220, 308, 228, 232, 118

Offset: 1

Vaclav Kotesovec, Jun 20 2020

Dirichlet convolution of A000203 with abs(A097945).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A000203, A001620, A097945, A176345, A326306.

Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))

Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Let f(s) = Product_{primes p} (1 - 1/(p^s + p)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)/2 + gamma - 3*zeta'(2)/Pi^2 - 1/4)*f(2) + f'(2)/2), where f(2) = A065463 = Product_{primes p} (1 - 1/(p*(p+1))) = 0.7044422009991655927366033503266372..., f'(2) = f(2) * Sum_{primes p} p*log(p) / ((p+1)*(p^2+p-1)) = 0.23219454323726621271960146689644280341444084188447499043209938838191022838..., for zeta'(2) see A073002 and gamma is the Euler-Mascheroni constant A001620.
a(n) = Sum_{d|n} A176345(d). - Ridouane Oudra, Jan 14 2022
Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - Amiram Eldar, Dec 25 2022

A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 2, 4, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The sum of these divisors is A365209(n).

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

Cf. A000005, A003586, A034444, A065330, A065331, A365209.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3, e + 1, 2]; 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,1] <= 3, f[i,2]+1, 2));}

Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000005(A065331(n)) * A034444(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).

A368547 Decimal expansion of the Wolf-Kawalec constant of index 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 30 2023

For the Wolf-Kawalec constant of index 0 see A368551.

For the Wolf-Kawalec constant of index 2 see A368568.

			0.23615288647712297486...

Artur Kawalec, On the series expansion of a square-free zeta series, arXiv:2312.16811 [math.NT], 2023.
Marek Wolf, Numerical Determination of a Certain Mathematical Constant Related to the Mobius Function, Computational Methods in Science and Technology, Volume 29 (1-4) 2023, 17-20 see formula (20).

Cf. A000796, A001620, A073002, A074962, A082633, A201994, A368551, A368568.

RealDigits[Limit[D[Zeta[x]/Zeta[2 x] - 6/(Pi^2 (x - 1)), x], x -> 1],
  10, 105][[1]]

Equals -(864*(zeta'(2))^2 - 72*Pi^2*(gamma*zeta'(2) + zeta''(2)) - 6*Pi^4*gamma_1)/Pi^6 where gamma_1 is A082633 negated.

Equals -(6*Pi^2*(2*(gamma + log(2) - 12*log(Glaisher) + log(Pi))*(gamma + 2*log(2) - 24*log(Glaisher) + 2*log(Pi)) - gamma_1) - 72*zeta''(2))/Pi^4 where Glaisher is the Glaisher-Kinkelin constant A (see A074962).

A368551 Decimal expansion of 6gamma/Pi^2 - 72zeta'(2)/Pi^4.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 29 2023

Also the Wolf-Kawalec constant of index 0.
For the Wolf-Kawalec constant of index 1 see A368547.
For the Wolf-Kawalec constant of index 2 see A368568.
Let g(n) be the Wolf-Kawalec constant of index n; then the function
  zeta(x)/zeta(2*x) - 6/(Pi^2*(x-1))
has the expansion
  Sum_{n>=0} (-1)^n*(g(n)/n!)*(x-1)^n
at x=1.

			1.0438945157119382974...

Artur Kawalec, On the series expansion of a square-free zeta series, arXiv:2312.16811 [math.NT], 2023.
Marek Wolf, Numerical Determination of a Certain Mathematical Constant Related to the Mobius Function, Computational Methods in Science and Technology, Volume 29 (1-4) 2023, 17-20 see formulas (26) and (27).

Cf. A000796, A001620, A008683, A073002, A074962, A368547, A368568.

RealDigits[6 EulerGamma/Pi^2 - 72 Zeta'[2]/Pi^4, 10, 105][[1]]

Equals (6/Pi^2)*(24*Glaisher - gamma - 2*log(2*Pi)) where Glaisher is A074962.

Equals lim_{x->oo} {(Sum_{n=1..x} abs(mu(n))/n) - 6*log(x)/Pi^2}.

A368568 Decimal expansion of the Wolf-Kawalec constant of index 2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 30 2023

For the Wolf-Kawalec constant of index 0 see A368551.

For the Wolf-Kawalec constant of index 1 see A368547.

			0.3193841204080142492494652...

Artur Kawalec, On the series expansion of a square-free zeta series, arXiv:2312.16811 [math.NT], 2023 see Table 1.
Marek Wolf, Numerical Determination of a Certain Mathematical Constant Related to the Mobius Function, Computational Methods in Science and Technology, Volume 29 (1-4) 2023, 17-20 see formula (20).

Cf. A001620, A073002, A086279, A201994, A368547, A368551.

RealDigits[Limit[D[D[Zeta[x]/Zeta[2 x] - 6/(Pi^2 (x - 1)), x], x], x -> 1],10,105][[1]]

Equals 6*(Pi^6*gamma_2 - 3456*(zeta'(2))^3 + 288*Pi^2*zeta'(2)*(gamma*zeta'(2) + 2*zeta''(2)) + 8*Pi^4*(3*gamma_1*zeta(2) - 3*gamma*zeta''(2) - 2*zeta'''(2)))/Pi^8 where gamma_2 is A086279.

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

cp's OEIS Frontend

A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

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A329117 Decimal expansion of Sum_{k>=1} (k^(1/k^2) - 1).

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A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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A252898 Decimal expansion of lim_{n->infinity} -FractionalPart[Zeta'(1+1/n)] or -FractionalPart[Zeta'(1-1/n)], where Zeta' is the first derivative of the Riemann zeta function.

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A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

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A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

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A368547 Decimal expansion of the Wolf-Kawalec constant of index 1.

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A368551 Decimal expansion of 6*gamma/Pi^2 - 72*zeta'(2)/Pi^4.

A368551 Decimal expansion of 6gamma/Pi^2 - 72zeta'(2)/Pi^4.