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%I #8 Dec 03 2018 21:36:44
%S 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,
%T 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,
%U 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
%N a(n) = round(1/frac(log(n!))) for n > 1; a(0) = a(1) = 0.
%C 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.
%e 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, ...
%e 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.
%o (PARI) A321991(n)=if(n>1,1\/frac(lngamma(n+1)),0)
%Y Cf. A025201, A046654.
%K nonn
%O 0,5
%A _M. F. Hasler_, Dec 03 2018
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