A117543 -id:A117543 - 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

A370093 Decimal expansion of Lichtman constant f(N*(2)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Feb 09 2024

Definition:
f(N*(k)) = Integral_{s>=1} P_k*(s), where P_k*(s) = Sum_{n>1 and (big) Omega(n)=k} mu(n)^2/n^s, where mu is Möbius (or Moebius) Mu function see A008683, and (big) Omega is number of prime divisors of n counted with multiplicity see A001222.
Lichtman constant f(N*(1)) see A137245.
Lichtman constant f(N*(2)) this sequence.
Lichtman constant f(N*(3)) see A370112.
Lichtman constant f(N*(4)) see A370113.
Limit_{k->oo} f(N*(k)) = 6/Pi^2 = 0.607927101854... see A059956.
Value computed and communicated by Bill Allombert.

			0.890925479476318332...

Bill Allombert, Results of pari computation of Lichtman constants f(N*(k)) with precision 500 decimals for k=1..20, email 20.06.2023.
Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019 (Figure 2 right column).

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370112, A370113.

pz(x)= sum(n=1,max(2,bitprecision(x)/x),my(a=moebius(n));if(a!=0,a*log(zeta(n*x))/n));
Lichtman(n)=intnum(s=1,[oo,log(2)],exp(-sum(i=1,n,pz(i*s)*x^i/i)+O(x^(n+1)))-1)
Lichtman(20)
\\ Bill Allombert, Feb 14 2014 [via Artur Jasinski]

A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 04 2008

The semiprime analog of A136141. To obtain the (smaller) sum over the squarefree semiprimes A006881, subtract the prime zeta functions of 4 ( A085964 ), 6, 8 etc. from this constant here. The first term in the representation as the geometric series in powers 1/q^s is in A117543 .

R. J. Mathar, Series of reciprocal powers of k-almost primes, constant B_{2,1} in table 8.

Equals 0.17105189297999663662220256437237421399124661203550059749107997... = 1/(4*3)+1/(6*5)+1/(9*8)+1/(10*9)+...

A372765 Decimal expansion of Lichtman constant f(N(2)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, May 14 2024

Definition:
f(N(k)) = Sum_{n>1 and (big) Omega(n)=k} 1/(n*log(n)), where (big) Omega is number of prime divisors of n counted with multiplicity see A001222.
f(N(k)) = Integral_{s>=1} P_k(s), where P_k(s) = Sum_{n>1 and (big) Omega(n)=k} 1/n^s.
Lichtman constant f(N(1)) see A137245.
Lichtman constant f(N(2)) this sequence.
Lichtman constant f(N(3)) see A372827.
Lichtman constant f(N(4)) see A372828.
Minimal value of f(N(k)) occurs for k=6 f(N(6)) = 0.9887534530145...
For k>=6, 1 > f(N(k+1)) > f(N(k)).
When k -> oo then f(N(k)) -> 1.
Value computed and communicated by Bill Allombert.

			1.1448165734059179915...

Bill Allombert, Results of pari computation of Lichtman constants f(N(k)) with precision 500 decimals for k=1..20, email 20.06.2023
Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019 (Figure 2 left column).

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372827, A372828.

pz(x)= sum(n=1, max(2, bitprecision(x)/x), my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
Lichtman(n)=intnum(s=1, [oo, log(2)], exp(sum(i=1, n, pz(i*s)*x^i/i)+O(x^(n+1)))-1)
Lichtman(20)
\\ Bill Allombert, May 14 2024 [via Artur Jasinski]

A131653 Decimal expansion of the sum of the reciprocals of squared 3-almost primes.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 10 2007, Mar 07 2008

zeta(2) = A013661 is 1 plus a sum over inverse squares of k-almost primes, k=1 to infinity, where A085548 represents k=1, A117543 represents k=2 and this constant here represents k=3.

			0.038516192982694640912837922628060395438900167478381571937...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009.

Cf. A000290, A013661, A014612, A085548, A085964, A085966, A117543, A085548.

RealDigits[PrimeZetaP[2]^3/6 + PrimeZetaP[6]/3 + PrimeZetaP[2]*PrimeZetaP[4]/2, 10, 120, -1][[1]] (* Amiram Eldar, Jun 25 2023 *)

Equals A085548^3 / 6 + A085966 / 3 + A085548 * A085964 / 2.

Equals Sum_{i>=1} 1/A000290(A014612(i)).

a(104) corrected by Amiram Eldar, Jun 25 2023

A372827 Decimal expansion of Lichtman constant f(N(3)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, May 14 2024

For definition and links see A372765.

			1.0308351017932175719...

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372765, A372828.

A372828 Decimal expansion of Lichtman constant f(N(4)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, May 14 2024

For definition and links see A372765.

			0.997342148595252359...

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372765, A372827.

A282468 Decimal expansion of the zeta function at 2 of every second prime number.

Original entry on oeis.org

1, 4, 4, 7, 1, 5, 5, 8, 6, 6, 8, 8, 7

Offset: 0

Terry D. Grant, Apr 14 2017

From Husnain Raza, Aug 30 2023: (Start)
Note that since p_n > n*log(n), we can place a bound on the tail of the sum:
Sum_{n >= N} (prime(2n))^(-2) <= Sum_{n >= N} (2*n*log(2n))^(-2) <= Integral_{x=N..oo} (2*x*log(2x))^(-2) dx.
Taking the sum over all primes < 10^12, we see that the constant lies between 0.14471558668870 and 0.14471558668873. (End)

			1/3^2 + 1/7^2 + 1/13^2 + 1/19^2 + 1/29^2 + ... = 0.14471558...

Husnain Raza, PARI program for calculating more digits.

Cf. A031215, A085548.

Zeta functions at 2: A085548 (for primes), A275647 (for nonprimes), A013661 (for natural numbers), A117543 (for semiprimes), A131653 (for triprimes), A222171 (for even numbers), A111003 (for odd numbers).

PARI
```
sum(n=1, 2500000, 1./prime(2*n)^2)
```
PARI
```
\\ see Raza link
```

Equals Sum_{n>=1} 1/A031215(n)^2 = Sum_{n>=1} 1/prime(2n)^2.

a(8)-a(12) from Husnain Raza, Aug 31 2023

A154943 Decimal expansion of the negated value of the sum_q [log(1-1/q)+1/q] over the semiprimes q.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

The semiprime analog of A143524. Taylor expansion of the logarithm shows that the value is sum_{s=2,3,..,infinity} P_2(s)/s, where P_2(s) are the semiprime zeta functions in Table 3 of the preprint arXiv:0803.0900. P_2(2)=A117543 and P_2(3)=0.023806..., P_2(4)=0.004994... etc.

			Equals 0.079848040306232691897402254...

Equals the negative of Sum_{i>=1} ( log(1-1/A001358(i)) +1/A001358(i)).

cp's OEIS Frontend

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

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A370093 Decimal expansion of Lichtman constant f(N*(2)).

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A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

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A372765 Decimal expansion of Lichtman constant f(N(2)).

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A131653 Decimal expansion of the sum of the reciprocals of squared 3-almost primes.

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A372827 Decimal expansion of Lichtman constant f(N(3)).

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A372828 Decimal expansion of Lichtman constant f(N(4)).

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A282468 Decimal expansion of the zeta function at 2 of every second prime number.

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