A222056 -id:A222056 - cp's OEIS frontend

A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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

Offset: 0

Cino Hilliard, Jul 03 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

			0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

Jason Kimberley, Table of n, a(n) for n = 0..1093
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Persi Diaconis, Frederick Mosteller, and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Lajos Hajdu and Rob Tijdeman, Integers represented by Lucas sequences, arXiv:2408.04982 [math.NT], 2024. See p. 16.
Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Gerhard Niklasch and Pieter Moree, Some number-theoretical constants. [Cached copy]
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Hanson Smith, Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves, arXiv:1810.04809 [math.NT], 2018.
Hanson Smith, Ramification in Division Fields and Sporadic Points on Modular Curves, U. Conn. (2020).
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Prime Zeta Function.
Index to constants which are prime zeta sums {2,0,0}

Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
Cf. A000720, A001248, A013661, A078437, A242301.

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(2,173)*10^105)));
// Jason Kimberley, Dec 30 2016

RealDigits[PrimeZetaP[2], 10, 105][[1]]  (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)

recip2(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^2; ); print(v) }

eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)),9)
lm=lambertw(log(4)/eps())\log(4);
sum(k=1,lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

sumeulerrat(1/p,2) \\ Hugo Pfoertner, Feb 03 2020

P(2) = Sum_{p prime} 1/p^2 = Sum_{n>=1} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020
Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - Amiram Eldar, Mar 12 2024

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Offset corrected by R. J. Mathar, Feb 05 2009

A253905 Decimal expansion of zeta(3)/zeta(2).

Original entry on oeis.org

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

Offset: 0

Geoffrey Critzer, Jan 18 2015

Three positive integers b, c, m are randomly selected (with replacement) from {1, 2, ..., n}. Let P(n) be the probability that the congruence b * x == c (mod m) has a solution. zeta(3)/zeta(2) is the limit of P(n) as n goes to infinity.

			0.73076296940143849872603673130771463952801160507937...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154.

Cf. A002117, A013661, A023900, A222056.

Drop[Flatten[RealDigits[N[Zeta[3]/Zeta[2], 75]]], -2]

zeta(3)/zeta(2) \\ Charles R Greathouse IV, Apr 20 2016

Equals A002117/A013661.
Equals Product_{p prime} (1 - 1/(p^2 + p + 1)). - Amiram Eldar, Jun 11 2023
Equals Sum_{k>=1} A023900(k)/k^3. - Amiram Eldar, Jan 25 2024

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

This is the density of A060687, the numbers with one 2 and the rest 1s in the exponents of its prime factorization. - Charles R Greathouse IV, Aug 03 2016

			0.200755722019265986996250723114404765853535555352561916...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.

Cf. A059956, A060687, A085548, A136141, A179119, A222056.

digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ Charles R Greathouse IV, Aug 03 2016

sumeulerrat(1/(p*(p+1)))/zeta(2) \\ Amiram Eldar, Mar 18 2021

Equals (6/Pi^2)*A179119.

A082293 Numbers having exactly one square divisor > 1.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 184, 188, 189, 198, 204, 207, 212

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by Peter Munn, Nov 17 2020]

The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - Amiram Eldar, Jul 07 2020

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

Cf. A005117, A046951, A222056.
Complement of A048111 within A013929.
Subsequence of A252849.
Disjoint union of A048109 and A060687.
A285508 is a subsequence.

Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* Amiram Eldar, Jul 07 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, primerange
def A082293(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x-sum(g(x//p**2) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A046951(a(n)) = 2.

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A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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A253905 Decimal expansion of zeta(3)/zeta(2).

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A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

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A082293 Numbers having exactly one square divisor > 1.

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