

A065027


a(n) is the smallest k > 0 such that n^k < k!.


8



1, 2, 4, 7, 9, 12, 14, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 79, 82, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 111, 114, 117, 119, 122, 125, 128, 130, 133, 136, 138, 141, 144, 147, 149, 152, 155, 157, 160, 163
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OFFSET

0,2


COMMENTS

Differences are 1, 2 or 3 (see A065067). The limit as n > infinity of a(n)/n is e.  Robert G. Wilson v, Dec 05 2001. [Apparently the Schonbek link contains a proof of the first assertion.]
a(10) = 25, a(100) = 269, a(1000) = 2714, a(10000) = 27177, a(10^5) = 271822, see A085830.
a(n) = least k such that geometric mean of {1,1/2,...,1/k} <= 1/n.  Clark Kimberling, Jul 11 2013
Let b(n) be the largest k such that n^k > k!. Then b(n) = a(n)1.  Joseph Damico, Jun 30 2019


LINKS

Harry J. Smith and Ely Golden, Table of n, a(n) for n = 0..10000 (first 1000 terms from Harry J. Smith)
Robert Israel, Plot of a(n)  n e + log(sqrt(2 Pi n)) for 1 <= n <= 20000
Tomas Schonbek, POLYA008: First n for which m^n < n! [From Nikos Apostolakis, Feb 17 2009]


FORMULA

It appears that L(n) < a(n)  n e + log(sqrt(2 Pi n)) < 1/2, where L(n) = 1/2 + o(1), and L(n) > 0.53 for all n.  Robert Israel, Oct 28 2016 (In other words, a(n)  n e + log(sqrt(2 Pi n)) < 1/2 for all n, and there is some function L(n) = 1/2 + o(1) such that 0.53 < L(n) < a(n)  n e + log(sqrt(2 Pi n)) for all n.  Charles R Greathouse IV, Nov 04 2016)


EXAMPLE

2^3 > 3! but 2^4 < 4!, so a(2)=4.


MAPLE

m:= 1:
for n from 1 to 100 do
while n^m >= m! do m:=m+1 od:
A[n]:= m;
od:
seq(A[n], n=0..100); # Robert Israel, Oct 28 2016


MATHEMATICA

Table[Length[Select[Table[m^n/n!, {n, 1, 180}], #>=1&]]+1, {m, 1, 61}]
sm0[n_]:=Module[{m=1}, While[n^m>=m!, m++]; m]; Array[sm0, 70] (* Harvey P. Dale, Jan 24 2018 *)


PROG

(PARI) { m=1; for (n=1, 1000, until (n^m < m!, m++); write("b065027.txt", n, " ", m) ) } \\ Harry J. Smith, Oct 03 2009


CROSSREFS

Sequence in context: A225000 A189677 A087733 * A165994 A163293 A188045
Adjacent sequences: A065024 A065025 A065026 * A065028 A065029 A065030


KEYWORD

easy,nonn


AUTHOR

Floor van Lamoen, Nov 02 2001


EXTENSIONS

More terms from Robert G. Wilson v, Dec 05 2001
a(0)=1 prepended by Alois P. Heinz, Dec 21 2019


STATUS

approved



