A136141 Decimal expansion of Sum_{p prime} 1/(p*(p-1)).

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

Offset: 0

Author

R. J. Mathar, Mar 09 2008

Keywords

• cons
• easy
• nonn

Comments

Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - Charles R Greathouse IV, Sep 06 2011

Sum of reciprocals of (proper) prime powers. The sum of reciprocals of all proper powers is A072102. - Charles R Greathouse IV, Apr 24 2012

Decimal expansion of the infinite sum of the reciprocals of the prime powers which are not prime (A246547). - Robert G. Wilson v, May 13 2019

See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - Robert G. Wilson v, May 13 2019

It easy to prove that this constant < 1 (Sum_{p prime} 1/(p*(p-1)) < Sum_{k>=2} 1/(k*(k-1)) = 1). Luthar (1969) asks for a better upper bound. The solution shows that this constant is < 3/2 - log(2) = 0.80685... . - Amiram Eldar, Feb 14 2025

Examples

			Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
= 0.7731566690497951278643674598559423956187413360831860483110060673567...
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
• Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 1002 terms from Jason Kimberley).
• Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint 1991.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• Stephan Ramon Garcia and Ethan Simpson Lee, Explicit conditional bounds for the residue of a Dedekind zeta-function at s = 1, arXiv:2506.17416 [math.NT], 2025. See p. 3.
• R. S. Luthar, Problem E 2192, Elementary Problems, The American Mathematical Monthly, Vol. 76, No. 8 (1969), p. 938; An Upper Bound, Solution to Problem E 2192, by O. P. Lossers, ibid., Vol. 77, No. 7 (1970), pp. 769-770.
• R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Tables 8 and 10.
• Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
• Index to constants which are prime zeta sums {1,1,0}

Crossrefs

Cf. A152447 (over the semiprimes), A000040, A000720, A001248, A046660 (excess, see first comment), A072102, A077761, A083342, A179119, A246547.

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).

Programs

Magma

R := RealField(105);
c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R,n)):n in[2..360]];
Reverse(IntegerToSequence(Floor(c*10^103))); // Jason Kimberley, Jan 12 2017

Mathematica

digits = 103; sp = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)

PARI

W(x)=solve(y=log(x)/2,max(1,log(x)),y*exp(y)-x)
eps()=2. >> (32*ceil(default(realprecision)/9.63))
primezeta(s)=my(t=s*log(2),iter=W(t/eps())\t);sum(k=1,iter, moebius(k)/k*log(abs(zeta(k*s))))
a(lim,e)={ \\ choose parameters to maximize speed and precision
    my(x,y=exp(W(lim)-.5));
    x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e,e*log(y));
    forprime(p=2,lim,x+=1/((p*1.)^e*(p-1)));
    x+sum(n=2,e,primezeta(n))
}; \\ Charles R Greathouse IV, Sep 07 2011

PARI

sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021

Formula

Equals Sum_{n>=1} 1/A036689(n).

Equals Sum_{s>=2} P(s), where P is the prime zeta function. - Charles R Greathouse IV, Sep 06 2011

Equals A083342 - A077761, that is, Sum_{n>=2} ((EulerPhi(n) - MoebiusMu(n))/n) * log(zeta(n)). - Jean-François Alcover, Sep 02 2015

Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - Amiram Eldar, Mar 12 2024

Extensions

More terms from D. S. McNeil, Sep 06 2011

More digits from Jean-François Alcover, Sep 02 2015