A060904 Largest power of 5 that divides n.

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

Keywords

• nonn
• easy
• mult

Comments

Also the largest power of 5 that divides the n-th Fibonacci number A000045(n).

Multiplicative with a(p^e) = 5^e if p = 5, else a(p^e) = 1. - Mitch Harris, Apr 19 2005

Also 5-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 5-adic absolute value. - Wolfdieter Lang, Jun 30 2014

Examples

			a(10) = 5 because 10 = 5 * 2.
		

References

• Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Crossrefs

Cf. A000045, A000203, A001620, A038500, A060865, A060901, A112765, A132739, A268354, A268357.

Programs

Magma

[5^Valuation(n,5): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

Maple

A060904 := n -> 5^padic[ordp](n,5): # Peter Luschny, Nov 26 2010

Mathematica

Table[5^IntegerExponent[n, 5], {n, 100}] (* Vincenzo Librandi, Dec 29 2015 *)

PARI

a(n)=5^valuation(n,5) \\ Charles R Greathouse IV, Aug 05 2015

Sage

[5^valuation(i,5) for i in [1..100]] # Tom Edgar, Mar 22 2014

Formula

If n is not divisible by 5, then a(n) = 1. If n = 5^k * m where m is not divisible by 5, then a(n) = 5^k.

Dirichlet g.f.: zeta(s)*(5^s-1)/(5^s-5). - R. J. Mathar, Jul 12 2012

a(n) = 5^A112765(n). - Tom Edgar, Mar 22 2014

From Peter Bala, Feb 21 2019: (Start)

a(n) = gcd(n,5^n).

a(n) = n/A132739(n).

O.g.f.: x/(1 - x) + 4*Sum_{n >= 1} 5^(n-1)*x^(5^n)/ (1 - x^(5^n)). (End).

a(n) = (1/5)*(sigma(5*n) - sigma(n))/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022

Sum_{k=1..n} a(k) ~ (4/(5*log(5)))*n*log(n) + (3/5 + 4*(gamma-1)/(5*log(5)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

Edited by Joerg Arndt and M. F. Hasler, Dec 29 2015