cp's OEIS Frontend

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

%I A088346 #25 Mar 31 2012 10:28:52
%S A088346 5,9,13,17,22,27,31,36,41,46,52,57,62,68,73,79,85,90,96,102,108,114,
%T A088346 120,126,132,138,145,151,157,164,170,176,183,189,196,202,209,215,222,
%U A088346 229,235,242,249,255,262,269,276,283,289,296,303,310,317,324,331,338,345,352
%N A088346 Smallest integer k for which exp(x) > x^n,  for all x>=k, n>=3.
%C A088346 n=3 is the starting index since exp(x) > x^n for all x>=0 when n=1,2.
%C A088346 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).
%t A088346 a[n_] := Ceiling[E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]
%t A088346 (* Also, the following code is from another definition of the *)
%t A088346 (* same sequence. *)
%t A088346 (* asymptotic prime like product function*) p[n_]=n!/(2*Product[Floor[i*Log[i]], {i, 2, Floor[n/Log[n]]}])
%t A088346 a0=Table[Floor[p[n]/p[n-1]], {n, 3, 500}];
%t A088346 (* composite like distribution*) Delete[Union[a0], 1];
%t A088346 (* pick of prime like numbers *) c=Table[If[a0[[n]]==1, n+2, 0], {n, 1, digits-3}];
%t A088346 d=Delete[Union[c], 1]
%Y A088346 Cf. A190951 (Closest integer to the largest real x such that exp(x) = x^n)
%Y A088346 Cf. A190952 (Largest integer k for which exp(k) < k^n)
%K A088346 nonn
%O A088346 3,1
%A A088346 _Roger L. Bagula_, Nov 07 2003
%E A088346 Partially edited _Charles R Greathouse IV_, Nov 02 2009
%E A088346 Provided new name, and added 2 initial terms, by _Shel Kaphan_, May 20 2011
%E A088346 Added Mathematica function, by _Shel Kaphan_, May 23 2011
%E A088346 Reverted to starting at n=3, improved Mathematica code, by _Shel Kaphan_, May 24 2011