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%I #27 Dec 25 2023 01:51:33
%S 1,1,3,2,5,1,7,4,9,5,11,2,13,7,15,8,17,3,19,10,21,11,23,1,25,13,27,14,
%T 29,5,31,16,33,17,35,6,37,19,39,20,41,7,43,22,45,23,47,2,49,25,51,26,
%U 53,9,55,28,57,29,59,10,61,31,63,32,65,11,67,34,69,35,71
%N Smallest integer of the form n/k!.
%C Equivalently, n divided by the largest factorial divisor of n.
%C Also, the smallest r such that n/r is a factorial number.
%C Positions of 1's are the factorial numbers A000142. Is every positive integer in this sequence? - _Gus Wiseman_, May 15 2019
%C Let m = A055874(n), the largest integer such that 1,2,...,m divides n. Then a(n*m!) = n since m+1 does not divide n, showing that every integer is part of the sequence. - _Etienne Dupuis_, Sep 19 2020
%H David A. Corneth, <a href="/A076934/b076934.txt">Table of n, a(n) for n = 1..10000</a>
%F From _Amiram Eldar_, Dec 25 2023: (Start)
%F a(n) = n/A055881(n)!.
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = BesselI(2, 2) = 0.688948... (A229020). (End)
%t Table[n/Max@@Intersection[Divisors[n],Array[Factorial,n]],{n,100}] (* _Gus Wiseman_, May 15 2019 *)
%t a[n_] := Module[{k=1}, While[Divisible[n, k!], k++]; n/(k-1)!]; Array[a, 100] (* _Amiram Eldar_, Dec 25 2023 *)
%o (PARI) first(n) = {my(res = [1..n]); for(i = 2, oo, k = i!; if(k <= n, for(j = 1, n\k, res[j*k] = j ) , return(res) ) ) } \\ _David A. Corneth_, Sep 19 2020
%Y Cf. A000142, A011371, A022559, A055874, A055881, A071626, A111701, A115627, A135291, A229020.
%Y Cf. A325272, A325273, A325508, A325509, A325543, A325544.
%K nonn,easy
%O 1,3
%A _Amarnath Murthy_, Oct 19 2002
%E More terms from _David A. Corneth_, Sep 19 2020
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