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%I #41 Mar 08 2024 05:42:31
%S 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,
%T 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,
%U 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
%N Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2)))
%C Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
%C Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
%C Numerators of partial products are A062271. Denominators are A062270.
%C 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.
%C 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
%C 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
%C Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - _Amiram Eldar_, Jun 20 2021
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
%D Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
%D Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.
%H Michel Balazard, Hubert Delange and Jean-Louis Nicolas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k57285043/f15.image">Sur le nombre de facteurs premiers des entiers</a>, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
%H Hubert Delange, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa19/aa1921.pdf">Sur des formules de Atle Selberg</a>, Acta Arith., Vol. 19 (1971), pp. 105-146.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, Sec. 2.1.
%H Emil Grosswald, <a href="http://doi.org/10.1215/S0012-7094-56-02305-5">The average order of an arithmetic function</a>, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
%H Jean-Louis Nicolas, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa44/aa4431.pdf">Sur la distribution des nombres entiers ayant une quantite fixee de facteurs premiers</a>, Acta Arith., Vol. 44 (1984), pp. 191-200.
%H Alfred Rényi, <a href="https://eudml.org/doc/257212">On the density of certain sequences of integers</a>, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
%F Equals 1/A005597.
%F 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) ....
%e Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577185338123...
%t 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 *)
%t $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 *)
%o (PARI) prodeulerrat((1 + 1/(p*(p-2))),,3) \\ _Hugo Pfoertner_, Aug 08 2020
%Y Cf. A005597.
%Y Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).
%K cons,nonn
%O 1,2
%A _Jonathan Sondow_, Nov 13 2009, Nov 17 2009
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