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%I #31 Jan 03 2023 09:22:02
%S 1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,
%T 1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,
%U 1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,121,1
%N Highest power of 11 dividing n.
%C The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 11.
%C This first index where this differs from A109014 is 121; a(121) = 121 and A109014(121) = 11.
%H Amiram Eldar, <a href="/A268357/b268357.txt">Table of n, a(n) for n = 1..10000</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) = 11^valuation(n,11).
%F Completely multiplicative with a(11) = 11, a(p) = 1 for prime p and p<>11. - _Andrew Howroyd_, Jul 20 2018
%F From _Peter Bala_, Feb 21 2019: (Start)
%F a(n) = gcd(n,11^n).
%F O.g.f.: x/(1 - x) + 10*Sum_{n >= 1} 11^(n-1)*x^(11^n)/ (1 - x^(11^n)). (End)
%F Sum_{k=1..n} a(k) ~ (10/(11*log(11)))*n*log(n) + (6/11 + 10*(gamma-1)/(11*log(11)))*n, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Nov 15 2022
%F Dirichlet g.f.: zeta(s)*(11^s-1)/(11^s-11). - _Amiram Eldar_, Jan 03 2023
%e Since 22 = 11 * 2, a(22) = 11. Likewise, since 11 does not divide 21, a(21) = 1.
%t Table[11^IntegerExponent[n, 11], {n, 130}] (* _Bruno Berselli_, Feb 03 2016 *)
%o (Sage) [11^valuation(i, 11) for i in [1..130]]
%o (Magma) [11^Valuation(n,11): n in [1..130]]; // _Vincenzo Librandi_, Feb 03 2016
%Y Cf. A001620, A006519, A038500, A234957, A234959, A109014, A268354, A060904.
%K nonn,easy,mult
%O 1,11
%A _Tom Edgar_, Feb 02 2016
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