A048684 - OEIS

%I #13 Sep 18 2024 08:43:51

%S 2,1,2,1,2,2,2,1,4,2,2,1,2,2,2,1,2,1,2,2,2,2,2,1,2,2,2,2,2,2,2,1,4,2,

%T 2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,

%U 2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2

%N Multiplicity of the maximum squarefree kernel function applied to the binomial coefficients C(n,k).

%H Amiram Eldar, <a href="/A048684/b048684.txt">Table of n, a(n) for n = 1..10000</a>

%e For n = 8, 9 or 10 the spectra of squarefree maximal divisors are {1,2,14,14,70,14,14,2,1}, {1,3,6,42,42,42,42,6,3,1} and {1,10,15,30,210,42,30,15,10,1}, respectively. The maxima (70,42 and 210) occur 1, 4 or 4 times. So a(8) = 1, a(9) = 4 and a(10) = 2.

%t rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; a[n_] := Module[{r = rad /@ Table[Binomial[n, k], {k, 0, n}]}, Count[r, Max[r]]]; Array[a, 100] (* _Amiram Eldar_, Sep 17 2024 *)

%o (PARI) rad(n) = vecprod(factor(n)[, 1]);

%o a(n) = {my(r = vector(n+1, k, rad(binomial(n,k-1))), rm = vecmax(r)); #select(x -> x==rm, r);} \\ _Amiram Eldar_, Sep 17 2024

%Y Cf. A007947, A020233, A020738, A001221, A001222, A000005.

%K nonn

%O 1,1

%A _Labos Elemer_