A269023 Complement of A269020: numbers not of the form ceiling(n^(1+1/n)).

2, 4, 8, 19, 51, 141, 392, 1079, 2957, 8072, 21987, 59825, 162695, 442342, 1202521, 3268920, 8885999, 24154826, 65659826, 178482140

Offset: 1

Author

Bob Selcoe, Feb 18 2016

Keywords

• nonn
• more

Comments

The limiting ratio is e (see comment in A059921).

Examples

			The term 8 appears because A269020(5)=7 and A269020(6)=9.
		

Crossrefs

Cf. A059921, A269020.

Programs

Mathematica

Complement[Range[1, 100000], Table[Ceiling[n^(1 + 1/n)], {n, 100000}]] (* Vaclav Kotesovec, Mar 12 2016 *)

PARI

a269020(n) = ceil(n^(1+1/n))
for(n=1, 1e20, if(a269020(n+1)-a269020(n) > 1, print1(a269020(n)+1, ", "))) \\ Felix Fröhlich, Mar 12 2016

Python

from itertools import count
def A269023(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        if x==1: return n+1
        z = x**x
        for y in count(x,-1):
            if y**(y+1) <= z:
                return n+y
            z //= x
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Extensions

a(18)-a(20) from Felix Fröhlich, Mar 12 2016