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
KEYWORD
nonn,easy,mult
AUTHOR
Tom Edgar, Feb 02 2016
EXTENSIONS
More terms from Antti Karttunen, Dec 22 2017
STATUS
approved