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A381456 Decimal expansion of Product_{p prime} p^(1/(p^2-1)).

%I A381456 #53 Apr 28 2025 00:05:12
%S A381456 1,7,6,8,1,9,8,0,7,8,1,5,3,2,4,4,9,8,4,1,3,0,8,5,3,0,7,7,2,3,1,4,9,6,
%T A381456 5,5,2,3,1,2,9,4,2,2,8,5,9,1,2,5,8,9,7,6,1,2,5,3,0,1,4,1,3,7,5,8,6,1,
%U A381456 0,7,9,1,4,6,0,0,0,0,4,3,0,0,9,3,0,3,1,5,7,1,7,1,0,7,2,8,5,1,5,6,1,9,3,8,0,6,6,6
%N A381456 Decimal expansion of Product_{p prime} p^(1/(p^2-1)).
%C A381456 The geometric mean of the zeta distribution with parameter value 2 (A381522) approaches this constant.
%C A381456 In general, for parameter value `s` it approaches e^(-zeta'(s)/zeta(s)). - _Jwalin Bhatt_, Feb 26 2025
%F A381456 Equals Product_{p>=2} p^(1/(p^2-1)) where p is prime.
%F A381456 Equals (A^12)/(2*Pi*(e^gamma)) where A = A074962 is the Glaisher-Kinkelin constant and gamma = A001620 is the Euler-Mascheroni constant.
%F A381456 Equals e^(-zeta'(2)/zeta(2)).
%F A381456 Equals exp((Sum_{k>=2} log(k)/(k^2))*(6/(Pi^2))).
%F A381456 Equals (Product_{k>=2} k^(1/(k^2)))^(6/(Pi^2)).
%F A381456 Equals exp(A306016). - _Hugo Pfoertner_, Feb 24 2025
%e A381456 1.768198078153244984130853077...
%t A381456 N[Exp[-Zeta'[2]/Zeta[2]], 120]
%o A381456 (Sage) N(exp(-diff(zeta(s:=var('s')), s).subs(s==2) / zeta(2)), 120)
%o A381456 (PARI) exp(-zeta'(2)/zeta(2)) \\ _Amiram Eldar_, Feb 24 2025
%o A381456 (Python)
%o A381456 from mpmath import zeta, diff, exp, mp
%o A381456 mp.dps = 120
%o A381456 const = exp(-diff(zeta, 2)/zeta(2))
%o A381456 A381456 = [int(d) for d in mp.nstr(const, n=mp.dps)[:-1] if d != '.']  # _Jwalin Bhatt_, Apr 08 2025
%Y A381456 Cf. A000040, A001620, A074962, A013661, A085548, A306016, A381522, A381898.
%K A381456 cons,nonn
%O A381456 1,2
%A A381456 _Jwalin Bhatt_, Feb 24 2025