A007845 Least positive integer k for which 5^n divides k!.

1, 5, 10, 15, 20, 25, 25, 30, 35, 40, 45, 50, 50, 55, 60, 65, 70, 75, 75, 80, 85, 90, 95, 100, 100, 105, 110, 115, 120, 125, 125, 125, 130, 135, 140, 145, 150, 150, 155, 160, 165, 170, 175, 175, 180, 185, 190, 195, 200, 200, 205, 210, 215, 220, 225, 225, 230, 235, 240, 245

Offset: 0

Author

Bruce Dearden and Jerry Metzger

Keywords

• nonn

Comments

Also the smallest factorial having at least n trailing zeros. - Jud McCranie, Oct 05 2010

a(n) ~ 4n, a(n) > 4n. Every positive multiple of 5 occurs as much as the exponent of 5 in the prime factorization. - David A. Corneth, Jul 12 2016

Least k such that A027868(k) >= n. - Robert Israel, Jul 12 2016

See A007843 and A007844 for the analog for 2 and 3 instead of 5. - M. F. Hasler, Dec 27 2019

References

• H. Ibstedt, Smarandache Primitive Numbers, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 216-229.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A007843, A007844, A027868.

Programs

Maple

1, seq(t $ padic:-ordp(t,5), t=5..1000, 5); # Robert Israel, Jul 12 2016

Mathematica

lpi[n_]:=Module[{k=1,n5=5^n},While[!Divisible[k!,n5],k++];k]; Array[ lpi,60,0] (* Harvey P. Dale, Jun 19 2012 *)

PARI

a(n) = {k = 1; while (valuation(k!, 5) < n, k++); k;} \\ Michel Marcus, Aug 19 2013

PARI

a(n) = {my(ck = 4 * n, k = 5 * floor(ck/5), t = 0); if(ck > 0, t = sum(i = 1, logint(ck,5),ck\=5)); while(t < n, k+=5; t+=valuation(k,5));max(1,k)} \\ David A. Corneth, Jul 12 2016

Formula

a(n) = 5*A228297(n) for n > 0: see A007843. - M. F. Hasler, Dec 27 2019