A074198 Largest k such that 1!*2!*3!*...*k! divides n!.

1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) is asymptotic to sqrt(2*n) and for n > 300 a(n) = ceiling(1/2 + sqrt(1+8*n)/2) (+0 or +1).

Example

18!/Product_{i=1..7} i! = 51051 but 18!/Product_{i=1..8} i! = 2431/1920, hence a(18) = 7.

Mathematica

a[n_] := Module[{k = 1}, NestWhile[#/(++k)! &, n!, IntegerQ]; k - 1]; Array[a, 100] (* Amiram Eldar, May 14 2024 *)

Prog

(PARI) a(n)=if(n<0, 0, my(s=1); while(n!%prod(i=1, s, i!) == 0, s++); s-1)

Crossrefs

Cf. A000178, A088302, A074199.

Sequence in context: A321318 A270362 A196383 * A196169 A330561 A048688

Adjacent sequences: A074195 A074196 A074197 * A074199 A074200 A074201

Keyword

easy,nonn

Author

Benoit Cloitre, Sep 17 2002

Status

approved