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%I #38 Jan 03 2023 02:31:30
%S 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,
%T 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,
%U 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
%N Highest power of 7 dividing n.
%C The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 7.
%H Antti Karttunen, <a href="/A268354/b268354.txt">Table of n, a(n) for n = 1..20790</a>
%H Tyler Ball, Tom Edgar, and Daniel Juda, <a href="http://dx.doi.org/10.4169/math.mag.87.2.135">Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem</a>, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
%F a(n) = 7^valuation(n,7).
%F a(n) = 7^A214411(n).
%F Completely multiplicative with a(7) = 7, a(p) = 1 for prime p and p <> 7. - _Andrew Howroyd_, Jul 20 2018
%F From _Peter Bala_, Feb 21 2019: (Start)
%F a(n) = gcd(n,7^n).
%F a(n) = n/A242603(n).
%F O.g.f.: x/(1 - x) + 6*Sum_{n >= 1} 7^(n-1)*x^(7^n)/ (1 - x^(7^n)). (End)
%F 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
%F Dirichlet g.f.: zeta(s)*(7^s-1)/(7^s-7). - _Amiram Eldar_, Jan 03 2023
%e Since 14 = 7 * 2, a(14) = 7. Likewise, since 7 does not divide 13, a(13) = 1.
%t 7^Table[IntegerExponent[n, 7], {n, 150}] (* _Vincenzo Librandi_, Feb 03 2016 *)
%o (Sage) [7^valuation(i, 7) for i in [1..100]]
%o (Magma) [7^Valuation(n,7): n in [1..150]]; // _Vincenzo Librandi_, Feb 03 2016
%o (PARI) a(n) = 7^valuation(n, 7) \\ _Michel Marcus_, Feb 05 2016
%Y Cf. A000420, A001620, A006519, A038500, A234957, A214411, A234959, A060904, A242603, A268357.
%K nonn,easy,mult
%O 1,7
%A _Tom Edgar_, Feb 02 2016
%E More terms from _Antti Karttunen_, Dec 22 2017
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