A020733 - OEIS

%I #28 Sep 21 2024 08:42:11

%S 2,1,2,1,2,5,4,1,4,2,4,1,2,5,8,1,2,5,8,2,6,7,8,5,8,11,2,2,4,11,10,3,8,

%T 2,6,3,6,2,4,1,2,5,8,2,12,16,16,5,6,13,8,12,12,4,8,5,4,5,6,4,2,6,10,1,

%U 2,7,6,5,2,2,12,15,16,2,8,11,2,10,10,11,2,6,12,3,16,2,4,8,10,5,2,2,4,6

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

%H Amiram Eldar, <a href="/A020733/b020733.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2000 from Robert Israel)

%e The number of distinct primes of binomial(15,k) are {0,2,3,3,4,4,4,4,4,4,4,4,3,3,2,0}. The maximum is 4 and it occurs 8 times, thus a(15) = 8.

%p f:= proc(n) local A,i;

%p   A:= [seq(nops(numtheory:-factorset(binomial(n,i))),i=0..n)];

%p   numboccur(max(A),A);

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, May 26 2020

%t a[n_] := Sort[Tally[Table[PrimeNu[Binomial[n, k]], {k, 0, n}]]][[-1, 2]];

%t Array[a, 100] (* _Jean-François Alcover_, Jun 09 2020 *)

%o (PARI) a(n) = {v = vector(n+1, k, omega(binomial(n, k-1))); m = vecmax(v); sum(i=1, n+1, v[i] == m);} \\ _Michel Marcus_, Dec 30 2013

%Y Cf. A001221, A048484, A048486.

%K nonn

%O 1,1

%A _Labos Elemer_