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A082908 Largest value of gcd(2^n, binomial(n,j)) with j=0..n-1; maximum value of largest power of 2 dividing binomial(n,j) in the n-th row of Pascal's triangle.

%I A082908 #34 Mar 15 2025 11:30:06
%S A082908 1,1,2,1,4,2,4,1,8,4,8,2,8,4,8,1,16,8,16,4,16,8,16,2,16,8,16,4,16,8,
%T A082908 16,1,32,16,32,8,32,16,32,4,32,16,32,8,32,16,32,2,32,16,32,8,32,16,32,
%U A082908 4,32,16,32,8,32,16,32,1,64,32,64,16,64,32,64,8,64,32,64,16,64,32,64,4,64,32
%N A082908 Largest value of gcd(2^n, binomial(n,j)) with j=0..n-1; maximum value of largest power of 2 dividing binomial(n,j) in the n-th row of Pascal's triangle.
%H A082908 Amiram Eldar, <a href="/A082908/b082908.txt">Table of n, a(n) for n = 0..10000</a>
%F A082908 a(n) = Max_{gcd(2^n, binomial(n, j)), j=0..n}.
%F A082908 a(n-1) = 2^floor(log_2(A000265(n))). - _Brad Clardy_, May 06 2013
%e A082908 For n = 10: the 10th row = {1,10,45,120,210,252,210,120,45,10,1}, the largest powers of 2 dividing the entries: {1,2,1,8,2,4,2,8,1,2,1}; maximum 2^k-divisor is a(10) = 8.
%t A082908 Table[Max[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}]], {n, 0, 128}]
%t A082908 a[n_] := 2^Floor[Log2[(n+1) / 2^IntegerExponent[n+1, 2]]]; Array[a, 82, 0] (* _Amiram Eldar_, Mar 15 2025 *)
%o A082908 (PARI) a(n)=n--; 2^(log(n>>valuation(n, 2)+.5)\log(2)) \\ _Charles R Greathouse IV_, May 06 2013
%Y A082908 Cf. A000005, A000079, A000265, A006519, A007318, A082907.
%K A082908 nonn
%O A082908 0,3
%A A082908 _Labos Elemer_, Apr 23 2003