A190952 -id:A190952 - cp's OEIS frontend

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

Roger L. Bagula, Nov 07 2003

nonn

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).

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)

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]

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

A190951 Closest integer to the largest real number x such that exp(x) = x^n, for n>=3.

Original entry on oeis.org

5, 9, 13, 17, 21, 26, 31, 36, 41, 46, 51, 56, 62, 67, 73, 79, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 157, 163, 169, 176, 182, 189, 195, 202, 208, 215, 221, 228, 235, 241, 248, 255, 262, 268, 275, 282, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352

Offset: 3

Shel Kaphan, May 23 2011

nonn

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

Cf. A088346 (Smallest integer k where exp(x)>x^n for all x>=k).

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

a[n_] := Round[ E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]

cp's OEIS Frontend

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

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