cp's OEIS Frontend

A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).

%I A375274 #8 Aug 09 2024 08:29:55
%S A375274 9,4,4,3,3,5,9,0,5,0,6,4,0,6,3,3,2,4,4,8,0,5,7,3,1,3,7,7,5,6,6,6,8,8,
%T A375274 0,5,6,1,4,6,3,4,5,8,3,2,2,2,0,2,3,5,5,5,9,2,3,6,8,3,7,7,0,4,5,5,9,3,
%U A375274 9,5,3,8,4,6,5,4,4,6,8,5,8,7,1,9,4,1,4,2,8,0,5,2,0,3,3,7,9,2,7,4,7,9,7,2,4
%N A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).
%C A375274 This constant was apparently first calculated by _Juan Arias-de-Reyna_ and _Peter J. C. Moses_ in 2015 (see A115063).
%F A375274 Equals Product_{p prime} (1 + Sum_{i>=2} (u(i) - u(i-1))/p^i), where u(i) = A010056(i) is the characteristic function of the Fibonacci numbers (A000045) (first formula at A115063).
%F A375274 Equals Product_{p prime} (1 + Sum_{i>=4} (-1)^(i+1)/p^A259623(i)).
%F A375274 Equals Product_{p prime} ((1 - 1/p) * (1 + Sum_{i>=2} 1/p^Fibonacci(i))).
%e A375274 0.94433590506406332448057313775666880561463458322202...
%t A375274 $MaxExtraPrecision = m = 500; em = 16; f[x_] := Log[(1 - x) * (1 + Sum[x^Fibonacci[e], {e, 2, em}])]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
%o A375274 (PARI) c(imax) = prodeulerrat((1-1/p)*(1 + sum(i = 2, imax, 1/p^fibonacci(i))));
%o A375274 f(prec) = {default(realprecision, prec); my(k = 2, c1 = 0, c2 = c(k)); while(c1 != c2, k++; c1 = c2; c2 = c(k)); c1;}
%o A375274 f(120)
%Y A375274 Cf. A000045, A010056, A115063, A259623.
%K A375274 nonn,cons
%O A375274 0,1
%A A375274 _Amiram Eldar_, Aug 09 2024