A321991 a(n) = round(1/frac(log(n!))) for n > 1; a(0) = a(1) = 0.

0, 0, 1, 1, 6, 1, 2, 2, 2, 1, 10, 2, 1, 2, 5, 1, 1, 2, 3, 3, 3, 3, 2, 2, 1, 277, 4, 2, 1, 4, 2, 11, 2, 18, 2, 7, 1, 3, 1, 2, 3, 29, 1, 2, 3, 8, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 1, 1, 107, 6, 3, 2, 1, 1, 5, 2, 1, 1, 4, 2, 1, 5, 2, 1, 3, 1, 15, 2, 1, 3, 1, 5, 1, 6, 2, 7

Offset: 0

Author

M. F. Hasler, Dec 03 2018

Keywords

• nonn

Comments

Motivated by Stirling's approximation s(n) = n*log(n) - n - log(2*Pi*n)/2 of log(n!), known to verify s(n) + 1/(12n+1) < log(n!) < s(n) + 1/12n. s(n) has the same integer part as log(n!) for all 1 < n < 10^6 at least, cf. A025201, but if the fractional part of log(n!) is less than 1/(12n+1), the approximation would yield the next lower integer. The first such n must have a(n) > 12n, so it is necessarily a record in this sequence, even a record of the sequence (a(n)/n). a(24) = 277 is a close miss, 12*24 = 288.

Examples

			Records occur at a(2) = 1, a(4) = 6, a(10) = 10, a(25) = 277, a(589) = 760, a(2965) = 921, a(3295) = 988, a(3802) = 1326, a(8743) = 1516, a(10634) = 2458, a(15404) = 11472, a(31672) = 56377, a(152170) = 162958, a(307001) = 295209, a(704236) = 491928, a(862929) = 528736, a(904492) = 1612903, a(1356678) = 5098244, ...
Among these, only a(2) = 1, a(4) = 6, and a(25) = 277 set a record for a(n)/n. No value is known for which a(n)/n >= 12.
		

Crossrefs

Cf. A025201, A046654.

Programs

PARI

A321991(n)=if(n>1,1\/frac(lngamma(n+1)),0)