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%I #33 May 20 2018 03:24:04
%S 14,22,33,42,44,56,57,59,74,107,113,115,1568,1571
%N Numbers k for which binomial(k, floor(k/2)) has the same number of unitary and non-unitary divisors.
%C Next term > 10^8. - _David A. Corneth_, May 14 2018
%C 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
%H David A. Corneth, <a href="/A048196/a048196_1.gp.txt">PARI link</a>
%F A034444(A001405(k)) = A048105(A001405(k)).
%e At k=59, the corresponding binomial coefficient, binomial(59,29) has 8192 divisors, of which 4096 are unitary and 4096 are not.
%o (PARI) isok(n) ={ n=binomial(n, floor(n/2)); sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); } \\ _Joerg Arndt_, May 13 2018
%o (PARI) \\ much faster:
%o isok(n) ={ n=binomial(n, floor(n/2)); my(u=1<<omega(n)); u==numdiv(n)-u; } \\ _Joerg Arndt_, May 13 2018
%o (PARI) \\ for a still faster program see the Corneth link.
%Y Cf. A001405, A034444, A034973, A048105, A048109.
%K nonn,hard
%O 1,1
%A _Labos Elemer_
%E a(9) .. a(14) from _Joerg Arndt_, May 13 2018
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