A055457 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2

Offset

1,5

Comments

More generally, consider the sequence defined by p^a(n) exactly divides p*n. For p = 3 we have A051064 and for p = 2 we have A001511.

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=5.)

Formula

G.f.: Sum_{k>=0} x^(5^k)/(1-x^5^k). - Ralf Stephan, Apr 12 2002

Multiplicative with a(p^e) = e+1 if p = 5, 1 otherwise.

a(n) = -Sum_{d|n} mu(5d)*tau(n/d). - Benoit Cloitre, Jun 21 2007

Dirichlet g.f.: zeta(s)/(1-1/5^s). - R. J. Mathar, Feb 09 2011

a(n) = A112765(5n). - R. J. Mathar, Jul 17 2012

a(5n) = 1 + a(n). a(5n+k) = 1 for k = 1..4. - Robert Israel, Dec 07 2015

G.f. satisfies A(x^5) = A(x) - x/(1-x). - Robert Israel, Dec 08 2015

a(n) = A112765(n) + 1. - Amiram Eldar, Sep 21 2020

Sum_{k=1..n} a(k) ~ 5*n/4. - Vaclav Kotesovec, Sep 21 2020

Example

a(5) = 2 since 5^2 exactly divides 5 times 5;
a(25) = 3 since 5^3 exactly divides 5 times 25;
a(125) = 4 since 5^4 exactly divides 5 times 125.

Maple

seq(padic:-ordp(5*n, 5), n=1..1000); # Robert Israel, Dec 07 2015

Mathematica

max = 1000; s = (1/x)*Sum[x^(5^k)/(1-x^5^k), {k, 0, Log[5, max] // Ceiling }] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 04 2015 *)
Table[IntegerExponent[n, 5] + 1, {n, 1, 100}] (* Amiram Eldar, Sep 21 2020 *)

Prog

(PARI) a(n)=-sumdiv(n, d, moebius(5*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007
(PARI) a(n)=valuation(5*n, 5) \\ Anders Hellström, Dec 04 2015

Crossrefs

Cf. A001511, A007949, A051064, A112765, A191610 (partial sums).

Sequence in context: A177706 A130782 A364358 * A277873 A032542 A107038

Adjacent sequences: A055454 A055455 A055456 * A055458 A055459 A055460

Keyword

nonn,mult,easy

Author

Alford Arnold, Jun 25 2000

Status

approved