A269020 -id:A269020 - cp's OEIS frontend

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

Bob Selcoe, Feb 18 2016

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

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

Cf. A059921, A269020.

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

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

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

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

A269024 a(n) = A269020(n) - n.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Bob Selcoe, Feb 18 2016

nonn

Same as subtracting the index (n) from the output (a(n)) in A269020.

			a(20)=4 because A269020(20) = 24, and 24-20=4.
a(5)=2, a(6)=3 and A269023(3)=8, so a(6) = 8-6+1 = 3.

Cf. A269020, A269023.

a(n) = either a(n-1) or a(n-1)+1.

When a(n) = a(n-1)+1, a(n) = A269023(a(n)) - n + 1.

cp's OEIS Frontend

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

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