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 A263922 Highest exponent in prime factorization of n-th central binomial coefficient. 4
 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 2, 2, 3, 2, 3, 3, 4, 2, 4, 3, 4, 4, 4, 4, 5, 2, 3, 4, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 2, 2, 3, 4, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 4, 3, 3, 4, 3, 4, 4, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) >= 2 for n > 4. a(n) is the maximum number of carries in base-p addition of n+n for primes p <= 2n. A000120(n) <= a(n) <= A070939(n). It appears that a(n) >= 3 for n > 1056. Any further n must be greater than 10^1000. Similarly it appears that a(n) >= 4 for n > 557056 and a(n) >= 5 for n > 1090519552. - Charles R Greathouse IV, Oct 31 2015 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI]. FORMULA a(n) = A051903(A000984(n)). EXAMPLE For n=3, C(6,3) = 20 = 2^2 * 5^1 so a(3) = 2. MAPLE f:= t -> max(seq(s[2], s=ifactors(t)[2])): seq(f(binomial(2*n, n)), n=1..100); # Robert Israel, Oct 29 2015 MATHEMATICA a[n_] := FactorInteger[Binomial[2 n, n]][[All, 2]] // Max; Array[a, 100] (* Jean-François Alcover, Nov 27 2015 *) PROG (PARI) f(n, p)=my(d=Vecrev(digits(n, p)), c); sum(i=1, #d, c=(2*d[i]+c>=p)) a(n)=my(r=hammingweight(n), L=sqrtnint(n, r+1), t); forprime(p=3, L, t=f(n, p); if(t>r, L=sqrtnint(n, 1+r=t)); if(p>=L, return(r))); r \\ Charles R Greathouse IV, Oct 29 2015 (PARI) vector(200, n, vecmax(factor(binomial(2*n, n))[, 2])) \\ Altug Alkan, Oct 30 2015 (Sage) max_exp = lambda n: max([p[1] for p in list(n.factor())]) print([max_exp(binomial(2*n, n)) for n in (1..87)]) # Peter Luschny, Oct 30 2015 def a(n):     N = 2*n     r = sum(N.digits(2))     b = 1+ZZ(N).nth_root(r, truncate_mode=1)[0]     for p in primes(3, b):         t, q = 0, N         while True:             q //= p             if q == 0: break             if (q & 1) == 1: t += 1         if t > r : r = t     return r print([a(n) for n in (1..87)]) # Peter Luschny, Nov 02 2015 (Haskell) a263922 = a051903 . a000984  -- Reinhard Zumkeller, Nov 19 2015 CROSSREFS Cf. A000120, A000984, A051903, A070939, A263924. Sequence in context: A064122 A323424 A334098 * A057526 A033265 A096004 Adjacent sequences:  A263919 A263920 A263921 * A263923 A263924 A263925 KEYWORD nonn,nice AUTHOR Robert Israel, Oct 29 2015 STATUS approved

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Last modified June 5 14:33 EDT 2020. Contains 334840 sequences. (Running on oeis4.)