prog.sf

#!/usr/bin/ruby

# a(n) = least k such that (k!*e^k)/(sqrt(2*Pi)*k^(k+1/2)) - 1 < 1/2^n.
# https://oeis.org/A247968

# Conjectured formula: a(n) = ceiling(2^n/12) (Charlie Neder, Mar 06 2019).

# Verified up to a(500) using binary search.
# Pseudo-verified up to a(10^4), by checking the formula for k = ceiling(2^n/12) and k = ceiling(2^n/12)-1.

#`( PARI/GP program:
    \p 1024
    isok(n,k) = (exp(log(1) - log(2)*n) > (exp((lngamma(k+1) + k) - (log(sqrt(2*Pi)) + log(k)*(k + 1/2))) - 1));
    a(n) = for(k=1, oo, if(isok(n, k), return(k)));
)

local Num!PREC = 2048

for n in (1..100) {

    var t = bsearch_min(1, 2**n, {|k|
        exp(log(1) - log(2)*n) <=> (exp((lngamma(k+1) + k) - (log(sqrt(2*Num.pi)) + log(k)*(k + 1/2))) - 1)
    })

    #~ var t = bsearch_min(1, 2**n, {|k|
        #~ (1 / 2**n) <=> ((gamma(k+1) * exp(k)) / (sqrt(2*Num.pi) * k**(k + 1/2)) - 1)
    #~ })

    assert_eq(t, ceil(2**n / 12))
    say "#{n} #{t}"
}

__END__
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