A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

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

Offset: 1

Author

Jonathan Sondow, Nov 13 2009, Nov 17 2009

Keywords

• cons
• nonn

Comments

Coefficient in formulas for the distribution of integers with a fixed number of prime factors.

Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.

Numerators of partial products are A062271. Denominators are A062270.

An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.

Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020

The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020

Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021

Examples

			Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577909192556...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
• Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
• Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.

Links

• Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
• Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
• Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
• Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
• Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixée de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
• Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Crossrefs

Cf. A005597.

Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).

Programs

Mathematica

s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)

PARI

prodeulerrat((1 + 1/(p*(p-2))),,3) \\ Hugo Pfoertner, Aug 08 2020

Formula

Equals 1/A005597.

Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....