A060904 - OEIS

%I #56 Nov 16 2022 06:55:25

%S 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,

%T 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,

%U 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

%N Largest power of 5 that divides n.

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

%C Multiplicative with a(p^e) = 5^e if p = 5, else a(p^e) = 1. - _Mitch Harris_, Apr 19 2005

%C 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

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

%H G. C. Greubel, <a href="/A060904/b060904.txt">Table of n, a(n) for n = 1..1000</a>

%H Tyler Ball, Tom Edgar, and Daniel Juda, <a href="http://dx.doi.org/10.4169/math.mag.87.2.135">Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem</a>, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

%F 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.

%F Dirichlet g.f.: zeta(s)*(5^s-1)/(5^s-5). - _R. J. Mathar_, Jul 12 2012

%F a(n) = 5^A112765(n). - _Tom Edgar_, Mar 22 2014

%F From _Peter Bala_, Feb 21 2019: (Start)

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

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

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

%F 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

%F 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

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

%p A060904 := n -> 5^padic[ordp](n,5): # _Peter Luschny_, Nov 26 2010

%t Table[5^IntegerExponent[n, 5], {n, 100}] (* _Vincenzo Librandi_, Dec 29 2015 *)

%o (Sage) [5^valuation(i,5) for i in [1..100]] # _Tom Edgar_, Mar 22 2014

%o (PARI) a(n)=5^valuation(n,5) \\ _Charles R Greathouse IV_, Aug 05 2015

%o (Magma) [5^Valuation(n,5): n in [1..100]]; // _Vincenzo Librandi_, Dec 29 2015

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

%K nonn,easy,mult

%O 1,5

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

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

%E Edited by _Joerg Arndt_ and _M. F. Hasler_, Dec 29 2015