A048569 -id:A048569 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Labos Elemer

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Cf. A000005, A048485, A048569, A048570.

f:= proc(n) local L,k;
  L:= [seq(numtheory:-tau(binomial(n,k)),k=0..n)];
  numboccur(max(L),L)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 17 2016

a[ n_] := If[ n < 1, 0, Last @ Last @ Tally @ Array[ Length @ Divisors @ Binomial[n, #] &, n+1, 0]]; (* Michael Somos, Nov 17 2016 *)

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

Original entry on oeis.org

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, 192, 256, 0, 256, 384, 0, 0, 0, 0, 0, 832, 768, 512, 0, 0, 0, 0, 384, 576, 1536, 3072, 2048, 8448, 7680, 5632, 0, 0, 0, 14336, 3584, 0, 0, 3072

Offset: 1

Labos Elemer

nonn

			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.

Cf. A000005, A048485, A048569.

020740 := proc(n)
    local a,k;
    a := -1 ;
    for k from 0 to n do
        a := max(a, numtheory[tau](binomial(n,k))) ;
    end do:
    a-numtheory[tau](binomial(n,floor(n/2))) ;
end proc:
seq(A020740(n),n=1..80); # R. J. Mathar, Nov 19 2024

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 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica