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 A000989 3-adic valuation of binomial(2*n, n): largest k such that 3^k divides binomial(2*n, n). 6
 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 0, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS a(n) = 0 if and only if n is in A005836. - Charles R Greathouse IV, May 19 2013 sign(a(n+1) - a(n)) is repeat [0, 1, -1]. - Filip Zaludek, Oct 29 2016 By Kummer's theorem, number of carries when adding n + n in base 3. - Robert Israel, Oct 30 2016 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..2000 Michael Gilleland, Some Self-Similar Integer Sequences. E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, Vol. 44 (1852), pp. 93-146; alternative link. Dorel Miheţ, Legendre's and Kummer's theorems again, Resonance, Vol. 15, No. 12 (2010), pp. 1111-1121; alternative link. Armin Straub, Victor H. Moll and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arithmetica, Vol. 140, No. 1 (2009), pp. 31-42. Wikipedia, Kummer's theorem. FORMULA a(n) = Sum_{k>=0} floor(2*n/3^k) - 2*Sum_{k>=0} floor(n/3^k). - Benoit Cloitre, Aug 26 2003 a(n) = A007949(A000984(n)). - Reinhard Zumkeller, Nov 19 2015 From Robert Israel, Oct 30 2016: (Start) If 2*n < 3^k then a(3^k+n) = a(n). If n < 3^k < 2*n then a(3^k+n) = a(n)+1. If n < 3^k then a(2*3^k+n) = a(n)+1. (End) a(n) = A053735(n) - A053735(2*n)/2. - Amiram Eldar, Feb 12 2021 MAPLE f:= proc(n) option remember; local k, m, d;    k:= floor(log[3](n));    d:= floor(n/3^k);    m:= n-d*3^k;    if d = 2 or 2*m > 3^k then procname(m)+1    else procname(m)    fi end proc: f(0):= 0: map(f, [\$0..100]); # Robert Israel, Oct 30 2016 MATHEMATICA p=3; Array[ If[ Mod[ bi=Binomial[ 2#, # ], p ]==0, Select[ FactorInteger[ bi ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 27*3, 0 ] Table[ IntegerExponent[ Binomial[2 n, n], 3], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2016 *) PROG (PARI) a(n) = valuation(binomial(2*n, n), 3) (PARI) a(n)=my(N=2*n, s); while(N\=3, s+=N); while(n\=3, s-=2*n); s \\ Charles R Greathouse IV, May 19 2013 (Haskell) a000989 = a007949 . a000984  -- Reinhard Zumkeller, Nov 19 2015 CROSSREFS Cf. A000984, A005836, A007949, A053735. Sequence in context: A062979 A114781 A083890 * A132401 A104273 A051778 Adjacent sequences:  A000986 A000987 A000988 * A000990 A000991 A000992 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified April 16 05:26 EDT 2021. Contains 343030 sequences. (Running on oeis4.)