A268354 Highest power of 7 dividing n.

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 7

Offset

1,7

Comments

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 7.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20790

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.

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Formula

a(n) = 7^valuation(n,7).

a(n) = 7^A214411(n).

Completely multiplicative with a(7) = 7, a(p) = 1 for prime p and p <> 7. - Andrew Howroyd, Jul 20 2018

From Peter Bala, Feb 21 2019: (Start)

a(n) = gcd(n,7^n).

a(n) = n/A242603(n).

O.g.f.: x/(1 - x) + 6*Sum_{n >= 1} 7^(n-1)*x^(7^n)/ (1 - x^(7^n)). (End)

Sum_{k=1..n} a(k) ~ (6/(7*log(7)))*n*log(n) + (4/7 + 6*(gamma-1)/(7*log(7)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022

Dirichlet g.f.: zeta(s)*(7^s-1)/(7^s-7). - Amiram Eldar, Jan 03 2023

Example

Since 14 = 7 * 2, a(14) = 7. Likewise, since 7 does not divide 13, a(13) = 1.

Mathematica

7^Table[IntegerExponent[n, 7], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)

Prog

(Sage) [7^valuation(i, 7) for i in [1..100]]
(Magma) [7^Valuation(n, 7): n in [1..150]]; // Vincenzo Librandi, Feb 03 2016
(PARI) a(n) = 7^valuation(n, 7) \\ Michel Marcus, Feb 05 2016

Crossrefs

Cf. A000420, A001620, A006519, A038500, A234957, A214411, A234959, A060904, A242603, A268357.

Sequence in context: A074465 A081229 A109010 * A117825 A010143 A101027

Adjacent sequences: A268351 A268352 A268353 * A268355 A268356 A268357

Keyword

nonn,easy,mult

Author

Tom Edgar, Feb 02 2016

Extensions

More terms from Antti Karttunen, Dec 22 2017

Status

approved