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A245285 a(n) is the smallest m such that m! > exp(n*m); or where the mean of the logs of the first m integers exceeds n. 0
2, 6, 18, 52, 145, 400, 1093, 2977, 8098, 22021, 59868, 162748, 442406, 1202597, 3269009, 8886102, 24154944, 65659960, 178482291, 485165185, 1318815724 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Rewriting the expression as log(m!)/m > n yields the interpretation: the smallest m such that the mean of the logs of the integers from 1 to m is greater than the integer n. This is also a useful way to determine a(n) for large n.

a(n) diverges from floor[e^(n+1)] slowly and almost linearly (to the low side) as n increases with these differences: 0, -1, -2, -2, -3, -3, -3, -3, -5, -5, -6, -6, -7, -7, -8, -8, -8, -9, -9, -10, -10.

The first five terms are the same as A128104(n-1).

From Jon E. Schoenfield, Dec 07 2014: (Start)

Let N = n+1, b = log(2*Pi)/2 - 1/4, and u = N/2 + b; then

a(n) = ceiling(x)

where x = exp(N - (u + 1/4) / e^N - (u^2 + 1/48) / e^(2*N))

is correct for all n in [0..10000], and probably for all nonnegative n. Rationale: let t be the real number that satisfies gamma(t+1) = exp(n*t); then the smallest integer m such that m! > exp(n*m) is ceiling(t). The above formula for x gives an approximation for t. At large values of n, x becomes huge, yet the approximation error x-t becomes extremely small, very roughly on the order of x^-2. E.g., at n=10000, x = 2.393...*10^4347, and x-t = 3.273...*10^-8676, so x agrees with t to more than 13000 significant digits. The above formula for a(n) will fail if and only if x and t fall on opposite sides of an integer, which seems extremely unlikely ever to occur.  (End)

LINKS

Table of n, a(n) for n=0..20.

FORMULA

No exact formula found so far.

EXAMPLE

For n = 0, 2!  >  exp(0*2) = 1.

For n = 1,  6! = 720  >  exp(1*6) = 403.

For n = 2,  18! = 6.402 * 10^15  >  exp(2*18) = 4.311 * 10^15.

PROG

(PARI) a(n)=m=1; while(m!<=exp(n*m), m++); m

n=1; while(n<15, print1(a(n), ", "); n++) \\ Derek Orr, Nov 13 2014

CROSSREFS

Cf. A128104.

Sequence in context: A192237 A034525 A018249 * A128104 A318570 A027059

Adjacent sequences:  A245282 A245283 A245284 * A245286 A245287 A245288

KEYWORD

nonn

AUTHOR

Richard R. Forberg, Nov 12 2014

STATUS

approved

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Last modified August 19 18:51 EDT 2019. Contains 326133 sequences. (Running on oeis4.)