A076934 Smallest integer of the form n/k!.

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, 29, 5, 31, 16, 33, 17, 35, 6, 37, 19, 39, 20, 41, 7, 43, 22, 45, 23, 47, 2, 49, 25, 51, 26, 53, 9, 55, 28, 57, 29, 59, 10, 61, 31, 63, 32, 65, 11, 67, 34, 69, 35, 71

Offset: 1

Author

Amarnath Murthy, Oct 19 2002

Keywords

• nonn
• easy

Comments

Equivalently, n divided by the largest factorial divisor of n.

Also, the smallest r such that n/r is a factorial number.

Positions of 1's are the factorial numbers A000142. Is every positive integer in this sequence? - Gus Wiseman, May 15 2019

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000142, A011371, A022559, A055874, A055881, A071626, A111701, A115627, A135291, A229020.

Cf. A325272, A325273, A325508, A325509, A325543, A325544.

Programs

Mathematica

Table[n/Max@@Intersection[Divisors[n],Array[Factorial,n]],{n,100}] (* Gus Wiseman, May 15 2019 *)
a[n_] := Module[{k=1}, While[Divisible[n, k!], k++]; n/(k-1)!]; Array[a, 100] (* Amiram Eldar, Dec 25 2023 *)

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

Formula

From Amiram Eldar, Dec 25 2023: (Start)

a(n) = n/A055881(n)!.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = BesselI(2, 2) = 0.688948... (A229020). (End)

Extensions

More terms from David A. Corneth, Sep 19 2020