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

1,1

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

Links

Table of n, a(n) for n=1..14.

David A. Corneth, PARI link

Formula

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

Example

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

Prog

(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<<omega(n)); u==numdiv(n)-u; } \\ Joerg Arndt, May 13 2018
(PARI) \\ for a still faster program see the Corneth link.

Crossrefs

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

Sequence in context: A306146 A318929 A187776 * A305084 A225710 A183185

Adjacent sequences: A048193 A048194 A048195 * A048197 A048198 A048199

Keyword

nonn,hard

Author

Labos Elemer

Extensions

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

Status

approved