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A020740 Max_{k=0..n} d(C(n,k)) - d(C(n,[ n/2 ])), where d() = number of divisors.

%I A020740 #10 Nov 19 2024 08:51:29
%S A020740 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,16,32,0,0,0,64,48,0,0,96,0,0,0,
%T A020740 192,256,0,256,384,0,0,0,0,0,832,768,512,0,0,0,0,384,576,1536,3072,
%U A020740 2048,8448,7680,5632,0,0,0,14336,3584,0,0,3072
%N A020740 Max_{k=0..n} d(C(n,k)) - d(C(n,[ n/2 ])), where d() = number of divisors.
%e A020740 n=20, d(C[ 20,10 ])= 48 and the d(C[ 20,k ]) values are 1,6,8,16,24,40,64,80. The maximum is 80, so the difference is 80-48 = 32.
%p A020740 020740 := proc(n)
%p A020740     local a,k;
%p A020740     a := -1 ;
%p A020740     for k from 0 to n do
%p A020740         a := max(a, numtheory[tau](binomial(n,k))) ;
%p A020740     end do:
%p A020740     a-numtheory[tau](binomial(n,floor(n/2))) ;
%p A020740 end proc:
%p A020740 seq(A020740(n),n=1..80); # _R. J. Mathar_, Nov 19 2024
%t A020740 Table[Max[Table[DivisorSigma[0,Binomial[n,k]],{k,0,n}]] - DivisorSigma[ 0,Binomial[n,Floor[n/2]]],{n,70}] (* _Harvey P. Dale_, Jul 18 2013 *)
%Y A020740 Cf. A000005, A048485, A048569.
%K A020740 nonn
%O A020740 1,18
%A A020740 _Labos Elemer_