A256392 -id:A256392 - cp's OEIS frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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

Offset: 0

Vaclav Kotesovec, Dec 17 2019

			0.5358961538283379998085026313185459506482223745141452711510108346133288119...

Cf. A055653, A062354, A065465, A175639, A256392, A332713.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 + p^3 - p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}]

(6/Pi^2) * prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Sep 08 2020

Equals (6/Pi^2) * A065465. - Amiram Eldar, Sep 08 2020

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A064549, A007913, A268335, A336643, A350390.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^e, p^(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,2]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 28 2020

a(n) is the least number k such that k*n (and also n/k) is an exponentially odd number (A268335). - Amiram Eldar, Nov 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A007913, A007947, A059841, A268335, A336644, A350390, A356191.

f[p_, e_] := p^(1 - Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336643(n) = (factorback(factorint(n)[, 1]) / core(n));

A336643(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(1-(f[i, 2]%2))));

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X^2) * (1 + X + p*X^2 - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336643(n): return prod(primefactors(n))//core(n) # Chai Wah Wu, Dec 30 2021

def A336643(n: int) -> int:
    return prod(b^(1 - e % 2) for (b, e) in list(factor(n)))
print([A336643(n) for n in range(1, 106)])  # Peter Luschny, Aug 23 2025

a(n) = A007947(n) / A007913(n).
Multiplicative with a(p^k) = p^(1-(k mod 2)) = p^A059841(k).
a(n) = n/A350390(n). - Amiram Eldar, Jan 01 2022
a(n) = A356191(n)/n. - Amiram Eldar, Nov 18 2022
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Sep 09 2023
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(1-5*s) + p^(2-5*s) + 2*p^(1-4*s) - p^(2-4*s) - p^(1-3*s) + p^(-3*s) - 2*p^(-2*s)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 12 * (log(n) + 3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.217778716619536378323007514119446813130797755001355937648276403523626491...,
f'(1) = f(1) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.716596820856763078660955244870812634072512131626849517007098664560806248...
and gamma is the Euler-Mascheroni constant A001620. (End)

A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

Equals (49/64)*(668/729)*(15304/15625)*(116724/117649)*... inserting p= A000040 = 2, 3, 5, 7.. into the factor. Slightly larger than Product_{p=primes} (1-3/p^3) = 0.534566872085103888416775...

			0.678234491917391978035...

Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 85.
Takashi Taniguchi, A mean value theorem for the square of class number times regulator of quadratic extensions, Annales de l'institut Fourier. Vol. 58, No. 2 (2008), pp. 625-670; arXiv preprint, arXiv:math/0410531 [math.NT], 2004-2006.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Taniguchi's Constant.
Wikipedia, Euler Product.

Cf. A256392, A330523.

read("transforms") : efact := 1-3/p^3+2/p^4+1/p^5-1/p^6 ; Digits := 130 : tm := 310 : subs (p=1/x,1/efact) ; taylor(%,x=0,tm) : L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:

digits = 105; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s];
LR = LinearRecurrence[{0, 0, 3, -2, -1, 1}, {0, 0, -9, 8, 5, -33}, 2 m0];
r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-3/p^3+2/p^4+1/p^5-1/p^6) \\ Amiram Eldar, Mar 04 2021

More digits from Jean-François Alcover, Apr 15 2016

A256391 a(n) = number of tuples (a,b,c,d) of natural numbers a,b,c,d <= n with gcd(a,b)=gcd(b,c)=gcd(c,d)=gcd(d,a)=1.

Original entry on oeis.org

1, 7, 35, 79, 243, 319, 787, 1155, 1859, 2295, 4267, 4891, 8295, 9743, 11851, 14539, 22191, 24359, 35427, 39387, 45915, 51687, 71171, 76407, 94911, 105047, 123251, 134447, 174003, 180835, 229783, 253007, 281447, 305111, 343315, 360215, 442547, 476115, 523111, 552307

Offset: 1

Juan Arias-de-Reyna, Mar 27 2015

nonn

The sequence has the asymptotics a(n) = rho*n^4 + O(n^3*log^2(n)) where rho=prod_p(1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.21777871661953... (product extended to primes). See A256392.

			For n=2, a(2)=7 counting the tuples (1,1,1,1), (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2), (2,1,2,1), (1,2,1,2).

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..100
J. Arias de Reyna and R. Heyman, Counting tuples restricted by pairwise primality, arXiv:1403.2769 [math.NT], 2014.
J. Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4

Cf. A256390.

A[M_] := A[M] = Module[{X, a1, a2, a3, a4, K, count, k},
    X = Flatten[
      Table[{a1, a2, a3, a4}, {a1, 1, M}, {a2, 1, M}, {a3, 1, M}, {a4,
         1, M}], 3];
    K = Length[X];
    count = 0;
    For[k = 1, k <= K, k++,
     {a1, a2, a3, a4} = X[[k]];
     If[(GCD[a1, a2] == 1) && (GCD[a2, a3] == 1) && (GCD[a3, a4] ==
         1) && (GCD[a4, a1] == 1), count = count + 1]];
    count];
Table[A[n], {n, 1, 20}]

a(n) = sum_a sum_b sum_c sum_d mu(a) mu(b) mu(c) mu(d) [n/gcd(a,b)][n/gcd(b,c)][n/gcd(c,d)][n/gcd(d,a)], where mu is Moebius function, a,b,c,d run through natural numbers.

A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

Original entry on oeis.org

1, 2, 8, 12, 32, 16, 72, 64, 108, 64, 200, 96, 288, 144, 256, 320, 512, 216, 648, 384, 576, 400, 968, 512, 1200, 576, 1296, 864, 1568, 512, 1800, 1536, 1600, 1024, 2304, 1296, 2592, 1296, 2304, 2048, 3200, 1152, 3528, 2400, 3456, 1936, 4232, 2560, 5292, 2400

Offset: 1

Jonathan Vos Post, May 27 2007

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A000010, A064840, A127473, A256392.

[ EulerPhi(n)*EulerPhi(n)*NumberOfDivisors(n) : n in [1..100] ];

Table[EulerPhi[n]^2 DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Dec 05 2012 *)

Multiplicative with a(p^e) = (e+1)*(p-1)^2*p^(2*e-2). - Amiram Eldar, Dec 29 2022
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(s-2)^2 * Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Sum_{k=1..n} a(k) ~ f(3) * n^3 * (log(n) + 2*gamma - 1/3 + f'(3)/f(3)) / 3, where
f(3) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264...,
f'(3) = f(3) * Sum_{p prime} 2*(2*p - 1) * log(p) / (p^3 + p^2 - 3*p + 1) = f(3) * 1.6860441157206199528397247528679297282000614932962665074593283751342385...
and gamma is the Euler-Mascheroni constant A001620. (End)

A196524 a(n) = phi(n)*tau(n^2).

Original entry on oeis.org

1, 3, 6, 10, 12, 18, 18, 28, 30, 36, 30, 60, 36, 54, 72, 72, 48, 90, 54, 120, 108, 90, 66, 168, 100, 108, 126, 180, 84, 216, 90, 176, 180, 144, 216, 300, 108, 162, 216, 336, 120, 324, 126, 300, 360, 198, 138, 432, 210, 300, 288, 360, 156, 378, 360, 504, 324, 252, 174, 720, 180, 270, 540, 416, 432, 540

Offset: 1

R. J. Mathar, Oct 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. M. Borwein, K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J., Vol. 7 (2003), pp. 95-127.
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Table[EulerPhi[n] DivisorSigma[0, n^2], {n, 70}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (2*e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=numdiv(n^2)*eulerphi(n) \\ Charles R Greathouse IV, Dec 07 2011

Multiplicative with a(p^e) = (2e+1)*(p-1)*p^(e-1), e>0.
a(n) = A048691(n)*A000010(n).
Dirichlet g.f.: zeta^3(s-1)*product_{primes p} (1-3/p^s -1/p^(2s-2) +4/p^(2s-1) -1/p^(3s-2)) = zeta^2(s-1)*product_{primes p} (1 +p^(1-s) +p^(1-2s) -3p^(-s)).
Sum_{k=1..n} a(k) ~ c * log(n)^2 * n^2 / 4, where c = A256392 = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.217778716619536378323007514119446813130797755... - Vaclav Kotesovec, Dec 18 2019
More precise asymptotics: Let f(s) = Product_{primes p} (1 - 3/p^s - 1/p^(2*s-2) + 4/p^(2*s-1) - 1/p^(3*s-2)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)^2/4 + (3*gamma/2 - 1/4)*log(n) + 3*gamma^2/2 - 3*gamma/4 - 3*sg1/2 + 1/8)*f(2) + (log(n)/2 + 3*gamma/2 - 1/4)*f'(2) + f''(2)/4), where f(2) = A256392, f'(2) = f(2) * Sum_{primes p} (5*p - 3) * log(p) / (p^3 + p^2 - 3*p + 1) = 0.44156369228425957720874599661015191553108775903124..., f''(2) = f'(2)^2/f(2) + f(2) * Sum_{primes p} = p*(7*p^3 - 2*p^2 - 5*p + 4) * log(p)^2 / (p^3 + p^2 - 3*p + 1)^2 = -0.0925787956842332743072787717877016487612772912975..., gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 18 2020

A275254 The bi-unitary gcd-sum function.

Original entry on oeis.org

1, 3, 5, 7, 9, 14, 13, 15, 17, 25, 21, 30, 25, 36, 43, 31, 33, 47, 37, 57, 61, 58, 45, 64, 49, 69, 53, 82, 57, 108, 61, 63, 99, 91, 113, 99, 73, 102, 117, 117, 81, 163, 85, 132, 141, 124, 93, 130, 97, 135, 155, 157, 105, 146, 181

Offset: 1

R. J. Mathar, Jul 21 2016

nonn

Row sums of A165430.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2, function P**(n).

Cf. A013661, A165430, A256392.

Pstarstar := proc(n)
    add(A165430(k,n),k=1..n) ;
end proc:

phi[x_, n_] := Sum[Boole[GCD[k, n] == 1], {k, 1, x}]; uphi[1]=1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); a[n_] := DivisorSum[n, uphi[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* Amiram Eldar, Sep 09 2019 *)

a(n) = Sum_{k=1..n} A165430(n,k).

Sum_{k=1..n} a(k) = c * n^2 * log(n) / 2 + O(n^2), where c = Product_{p prime} (1 - (3*p-1)/(p^2*(p+1))) = zeta(2) * Product_{p prime} (1 - (2*p-1)^2/p^4) = A013661 * A256392 = 0.35823163000196141456... . - Amiram Eldar, Dec 22 2023

A319592 Decimal expansion of the probability that an integer 4-tuple is pairwise coprime.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 27 2019

			0.114884044080228788729251276701599097848713552687283...

László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quarterly, Vol. 40, No. 1 (2002), pp. 13-18.

Cf. A059956, A065473, A256392.

$MaxExtraPrecision = 1000; nm = 1000; c = LinearRecurrence[{-2, 3}, {0, -12}, nm]; f[x_] := (1 - x)^3*(1 + 3*x); RealDigits[f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

prodeulerrat((1 - 1/p)^3 * (1 + 3/p)) \\ Amiram Eldar, Jun 29 2023

Equals Product_{p prime} (1 - 1/p)^3 * (1 + 3/p).

A354709 Decimal expansion of Sum_{p prime} 3(2p-1)log(p)/(p^3 + p^2 - 3p + 1).

Original entry on oeis.org

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

Offset: 1

David Nguyen, Jun 03 2022

Also logarithmic derivative of A(s,w) at (0,0), where A(s,w) = Product_{p prime} (1 - (1 - (p*(1 - p^(-1-s))^3)/(-1+p))*(1 - (p*(1 - p^(-1-w))^3)/(-1+p))), with A(0,0) = A256392.

			2.52906617358092992925958712930189459230009223994439976118899256270135780066...

Cf. A256392.

Block[{$MaxExtraPrecision = 1000},
Do[CC = Join[{0},
    Series[(3 (-1 + 2 p))/(1 - 3 p + p^2 + p^3) //. p -> 1/x, {x, 0,
       t}][[3]]];
  Print[N[-Sum[
        CC[[k]]*(PrimeZetaP'[k] + Log[2]/2^k), {k, 1, Length[CC]}] + (
      3 (-1 + 2 p) Log[p])/(1 - 3 p + p^2 + p^3) //. p -> 2, 75]], {t,
    1000, 1500, 100}]]
ratfun = 3*(2*p - 1)/(p^3 + p^2 - 3*p + 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 120]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 04 2022 *)

More digits from Vaclav Kotesovec, Jun 04 2022

cp's OEIS Frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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A356191 a(n) is the smallest exponentially odd number that is divisible by n.

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A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

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A175639 Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).

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Maple

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A256391 a(n) = number of tuples (a,b,c,d) of natural numbers a,b,c,d <= n with gcd(a,b)=gcd(b,c)=gcd(c,d)=gcd(d,a)=1.

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A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

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Magma

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Formula

A196524 a(n) = phi(n)*tau(n^2).

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Formula

A354709 Decimal expansion of Sum_{p prime} 3(2p-1)log(p)/(p^3 + p^2 - 3p + 1).