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A020738 Consider number of divisors of binomial(n, k), k=0..n; a(n) = multiplicity of the maximum value.

%I A020738 #27 Sep 21 2024 08:41:40
%S A020738 2,1,2,1,2,1,4,3,4,1,4,3,2,1,2,1,6,2,6,2,6,1,8,2,2,1,4,2,2,1,10,4,2,5,
%T A020738 2,2,2,1,2,1,6,2,2,2,4,1,2,1,2,2,6,2,4,2,2,4,2,1,10,2,2,3,4,8,2,2,2,5,
%U A020738 2,2,2,2,2,2,2,5,2,2,6,2,2,2,12,2,2,1,2,4,4,2,2,2,2,1,2,2,2,1,4,2,4,2
%N A020738 Consider number of divisors of binomial(n, k), k=0..n; a(n) = multiplicity of the maximum value.
%H A020738 Amiram Eldar, <a href="/A020738/b020738.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Robert Israel)
%e A020738 If n = 23, the numbers of divisors of {binomial(23, k)} are {1, 2, 4, 8, 16, 16, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 32, ...}. The maximum occurs 8 times, so a(23) = 8.
%p A020738 f:= proc(n) local L,k;
%p A020738   L:= [seq(numtheory:-tau(binomial(n,k)),k=0..n)];
%p A020738   numboccur(max(L),L)
%p A020738 end proc:
%p A020738 map(f, [$1..200]); # _Robert Israel_, Nov 17 2016
%t A020738 a[ n_] := If[ n < 1, 0, Last @ Last @ Tally @ Array[ Length @ Divisors @ Binomial[n, #] &, n+1, 0]]; (* _Michael Somos_, Nov 17 2016 *)
%Y A020738 Cf. A000005, A048485, A048569, A048570.
%K A020738 nonn
%O A020738 1,1
%A A020738 _Labos Elemer_