A039593 -id:A039593 - cp's OEIS frontend

A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 2, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jul 27 2000

nonn

Number of prime divisors of the largest square dividing A001405(n). (A prime divisor is nonunitary iff its exponent exceeds 1.)

			For n=10, binomial(10, 5) = 252 = 2*2*3*3*7 has 3 prime divisors of which only one, p=7, is unitary, while 2 and 3 are not. So a(10)=2.
For n=256, binomial(256, 128) also has only 2 prime divisors (3 and 13) whose exponents exceed 1 (4 and 2, respectively), thus a(256)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001405, A034444, A034973, A039593, A056057, A056173.

Table[Count[FactorInteger[Binomial[n, Floor[n/2]]][[All, -1]], e_ /; e > 1], {n, 105}] (* Michael De Vlieger, Mar 05 2017 *)

a(n)=omega(core(binomial(n, n\2), 1)[2]) \\ Charles R Greathouse IV, Mar 09 2017

a(n) = A001221(A000188(A001405(n))).

a(n) = A001221(A056057(n)).

Edited by Jon E. Schoenfield, Mar 05 2017

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)).

Original entry on oeis.org

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

Labos Elemer, Aug 10 2000

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A039593, A000005, A055231, A007913, A055229, A001405, A056673, A056674.

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 *)

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

a(n) = A039593(n) - A000005(A055231(x)) = A039593(n) - A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).
a(n) = A039593(n) - A056673(n). - Sean A. Irvine, May 02 2022
a(n) = A056674(A001405(n)). - Amiram Eldar, Oct 04 2024

cp's OEIS Frontend

A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

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