A088346 Smallest integer k for which exp(x) > x^n, for all x>=k, n>=3.

5, 9, 13, 17, 22, 27, 31, 36, 41, 46, 52, 57, 62, 68, 73, 79, 85, 90, 96, 102, 108, 114, 120, 126, 132, 138, 145, 151, 157, 164, 170, 176, 183, 189, 196, 202, 209, 215, 222, 229, 235, 242, 249, 255, 262, 269, 276, 283, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352

Offset

3,1

Comments

n=3 is the starting index since exp(x) > x^n for all x>=0 when n=1,2.

This function also cancels out a different set of numbers from the factorial than the primes using the asymptotic behavior of prime(n) and pi(n).

Links

Table of n, a(n) for n=3..60.

Mathematica

a[n_] := Ceiling[E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]
(* Also, the following code is from another definition of the *)
(* same sequence. *)
(* asymptotic prime like product function*) p[n_]=n!/(2*Product[Floor[i*Log[i]], {i, 2, Floor[n/Log[n]]}])
a0=Table[Floor[p[n]/p[n-1]], {n, 3, 500}];
(* composite like distribution*) Delete[Union[a0], 1];
(* pick of prime like numbers *) c=Table[If[a0[[n]]==1, n+2, 0], {n, 1, digits-3}];
d=Delete[Union[c], 1]

Crossrefs

Cf. A190951 (Closest integer to the largest real x such that exp(x) = x^n)

Cf. A190952 (Largest integer k for which exp(k) < k^n)

Sequence in context: A314684 A247128 A314685 * A314686 A314687 A314688

Adjacent sequences: A088343 A088344 A088345 * A088347 A088348 A088349

Keyword

nonn

Author

Roger L. Bagula, Nov 07 2003

Extensions

Partially edited Charles R Greathouse IV, Nov 02 2009

Provided new name, and added 2 initial terms, by Shel Kaphan, May 20 2011

Added Mathematica function, by Shel Kaphan, May 23 2011

Reverted to starting at n=3, improved Mathematica code, by Shel Kaphan, May 24 2011

Status

approved