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