A056676 Number of non-unitary but squarefree divisors of binomial(n,floor(n/2)). Also number of nonsquarefree but unitary divisors of binomial(n,floor(n/2)).
0, 0, 0, 0, 0, 2, 0, 0, 4, 6, 0, 8, 8, 8, 8, 16, 0, 16, 0, 16, 32, 32, 0, 32, 48, 48, 56, 56, 96, 96, 64, 128, 128, 192, 256, 384, 384, 384, 512, 768, 512, 512, 512, 512, 448, 448, 768, 896, 896, 896, 896, 896, 768, 768, 2048, 2048, 4096, 4096, 2048, 2048, 2048, 2048
Offset: 1
Keywords
Examples
For n = 14, binomial(14,7) = 3432 has 32 divisors, 16 unitary, 16 squarefree. The size of overlap is 8. The complementary parts are: non-unitary/squarefree set ={2,6,22,26,66,78,286,828}, while the unitary/not squarefree set of equal size is {8,24,88,104,264,312,1144,3432}. So a(14) = 8.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
Programs
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Mathematica
f[p_, e_] := If[e == 1, 2, 1]; a[1] = 0; a[n_] := 2^Length[fct = FactorInteger[Binomial[n, Floor[n/2]]]] - Times @@ f @@@ fct; Array[a, 100] (* Amiram Eldar, Oct 04 2024 *)
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PARI
a(n) = {my(f = factor(binomial(n, n\2)), e = f[, 2]); 2^omega(f) - prod(i = 1, #e, if(e[i] == 1, 2, 1)); } \\ Amiram Eldar, Oct 04 2024