A048196 Numbers k for which binomial(k, floor(k/2)) has the same number of unitary and non-unitary divisors.

14, 22, 33, 42, 44, 56, 57, 59, 74, 107, 113, 115, 1568, 1571

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• hard

Comments

Next term > 10^8. - David A. Corneth, May 14 2018

Numbers k where b = binomial(k, floor(k/2)) is of the form p_i ^ e_i where p_i is the i-th prime in the factorization of b, e_i = 1 except exactly one e_i = 3 for i > 1. - David A. Corneth, May 13 2018

Examples

			At k=59, the corresponding binomial coefficient, binomial(59,29) has 8192 divisors, of which 4096 are unitary and 4096 are not.
		

Links

• David A. Corneth, PARI link

Crossrefs

Cf. A001405, A034444, A034973, A048105, A048109.

Programs

PARI

isok(n) ={ n=binomial(n, floor(n/2)); sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); } \\ Joerg Arndt, May 13 2018

PARI

\\ much faster:
isok(n) ={ n=binomial(n, floor(n/2)); my(u=1<Joerg Arndt, May 13 2018

PARI

\\ for a still faster program see the Corneth link.

Formula

A034444(A001405(k)) = A048105(A001405(k)).

Extensions

a(9) .. a(14) from Joerg Arndt, May 13 2018