A214411 The maximum exponent k of 7 such that 7^k divides n.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Redjan Shabani, Jul 16 2012

Keywords

• nonn
• easy

Comments

7-adic valuation of n.

Examples

			n=147 = 3*7*7 is divisible by 7^2, so a(147)=2.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

Crossrefs

Cf. A007814 (2-adic), A007949 (3-adic), A112765 (5-adic), A082784.

Programs

Maple

seq(padic:-ordp(n,7), n=1..100); # Robert Israel, Mar 05 2020

Mathematica

mek[n_]:=Module[{k=Ceiling[Log[7,n]]},While[!Divisible[n,7^k],k--];k]; Array[ mek,140] (* Harvey P. Dale, Mar 27 2017 *)
IntegerExponent[Range[150],7] (* Suggested by Amiram Eldar *) (* Harvey P. Dale, Mar 07 2020 *)

PARI

a(n)=valuation(n,7) \\ Charles R Greathouse IV, Jul 17 2012

PARI

A=vector(1000);for(i=1,log(#A+.5)\log(7),forstep(j=7^i,#A,7^i,A[j]++));A \\ Charles R Greathouse IV, Jul 17 2012

Formula

G.f.: Sum_{k>=1} x^(7^k)/(1-x^(7^k)). See A112765. - Wolfdieter Lang, Jun 18 2014

If n == 0 (mod 7) then a(n) = 1 + a(n/7), otherwise a(n) = 0. - M. F. Hasler, Mar 05 2020

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/6. - Amiram Eldar, Jan 17 2022

a(n) = 7*Sum_{j=1..floor(log(n)/log(7))} frac(binomial(n, 7^j)*7^(j-1)/n). - Dario T. de Castro, Jul 12 2022