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

1,5

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

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.

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

Example

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

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 *)

Prog

(Sage) [5^valuation(i, 5) for i in [1..100]] # Tom Edgar, Mar 22 2014
(PARI) a(n)=5^valuation(n, 5) \\ Charles R Greathouse IV, Aug 05 2015
(Magma) [5^Valuation(n, 5): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

Crossrefs

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

Sequence in context: A257461 A129398 A109009 * A351084 A135469 A348735

Adjacent sequences: A060901 A060902 A060903 * A060905 A060906 A060907

Keyword

nonn,easy,mult

Author

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

Extensions

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

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

Status

approved