A375274 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

This constant was apparently first calculated by Juan Arias-de-Reyna and Peter J. C. Moses in 2015 (see A115063).

LINKS

Table of n, a(n) for n=0..104.

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))).

EXAMPLE

0.94433590506406332448057313775666880561463458322202...

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]]

PROG

(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)

CROSSREFS

Cf. A000045, A010056, A115063, A259623.

Sequence in context: A117018 A193712 A249600 * A262313 A097878 A173571

Adjacent sequences: A375271 A375272 A375273 * A375275 A375276 A375277

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Aug 09 2024

STATUS

approved