

A056673


Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the special squarefree part of A001405(n), A056060(n).


0



1, 2, 2, 4, 4, 2, 4, 8, 4, 2, 16, 8, 8, 8, 8, 16, 32, 16, 32, 16, 32, 32, 64, 32, 16, 16, 8, 8, 32, 32, 64, 128, 128, 64, 256, 128, 128, 128, 512, 256, 512, 512, 512, 512, 64, 64, 256, 128, 128, 128, 128, 128, 256, 256, 2048, 2048, 4096, 4096, 2048, 2048, 2048, 2048
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OFFSET

1,2


LINKS

Table of n, a(n) for n=1..62.


FORMULA

a(n) = A000005(A055231(x)) = A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).


EXAMPLE

n=14: binomial(15,7) = 3432 = 2*2*2*3*11*13, which has 32 divisors. Of those divisors, 16 are unitary: {1, 3, 8, 11, 13, 24, 33, 39, 88, 104, 143, 264, 312, 429, 1144, 3432}; 16 are squarefree: {1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858}. Only 8 of the divisors belong to both classes: {1, 3, 11, 13, 33, 39, 143, 429}. Thus, a(14)=8.


MATHEMATICA

Table[With[{m = Binomial[n, Floor[n/2]]}, DivisorSum[m, 1 &, And[CoprimeQ[#, m/#], SquareFreeQ@ #] &]], {n, 62}] (* Michael De Vlieger, Sep 05 2017 *)


PROG

(PARI) a(n) = my(b=binomial(n, n\2)); sumdiv(b, d, issquarefree(d) && (gcd(d, b/d) == 1)); \\ Michel Marcus, Sep 05 2017


CROSSREFS

Cf. A000005, A001405, A007913, A055229, A055231, A056060.
Sequence in context: A214516 A238004 A048244 * A128442 A290633 A038674
Adjacent sequences: A056670 A056671 A056672 * A056674 A056675 A056676


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 10 2000


STATUS

approved



