A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).
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, 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, 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
Offset: 0
Examples
0.94433590506406332448057313775666880561463458322202...
Programs
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Mathematica
$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]] -
PARI
c(imax) = prodeulerrat((1-1/p)*(1 + sum(i = 2, imax, 1/p^fibonacci(i)))); f(prec) = {default(realprecision, prec); my(k = 2, c1 = 0, c2 = c(k)); while(c1 != c2, k++; c1 = c2; c2 = c(k)); c1;} f(120)
Formula
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).
Equals Product_{p prime} (1 + Sum_{i>=4} (-1)^(i+1)/p^A259623(i)).
Equals Product_{p prime} ((1 - 1/p) * (1 + Sum_{i>=2} 1/p^Fibonacci(i))).
Comments