A380334 Decimal expansion of the smallest possible average gap for the prime number subsequence P" = A262275.
9, 4, 5, 1, 9, 3, 9, 8, 1, 3, 4, 4, 7, 8, 0, 6, 3, 1, 1, 4, 1, 0, 7, 5, 0, 2, 2, 4, 7, 0, 8, 4, 0, 4, 2, 1, 7, 5, 4, 2, 0, 4, 5, 1, 5, 5, 4, 8, 6, 7, 9, 6, 2, 6, 3, 0, 9, 1, 6, 8, 3, 2, 3, 6, 5, 0, 5, 4, 8, 7, 9, 5, 6, 5, 9, 6, 7, 6, 5, 2, 1, 0, 6, 1, 0, 0, 9, 7, 8, 7, 0, 9, 0, 8, 4, 1, 9, 6, 6
Offset: 1
Examples
9.45193981344780631141075022470840...
Programs
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Mathematica
f[x_] := Log[x!]*{Log[x!]+1}-x*{x-1}; root = FindRoot[f[x] == 0, {x, 3},WorkingPrecision -> 100]; y=root; z=x/. y; RealDigits[Log[z!]*{Log[z!]+1}] -
PARI
my(y=solve(x=3, 4, lngamma(x+1)-x+1)); y*(y-1) \\ Michel Marcus, Jan 28 2025
Formula
Equals x*(x-1) where x = A338460.