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Numbers k for which binomial(k, floor(k/2)) has fewer unitary than non-unitary divisors.
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%I #17 Oct 05 2024 09:07:17

%S 10,25,26,27,28,29,30,34,36,37,38,40,45,46,47,48,49,50,51,52,53,54,58,

%T 60,61,62,63,64,66,68,69,70,75,76,77,78,80,81,82,83,84,85,86,87,88,89,

%U 90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,108,109,110

%N Numbers k for which binomial(k, floor(k/2)) has fewer unitary than non-unitary divisors.

%C A048111 applied to central binomial coefficients.

%H Amiram Eldar, <a href="/A048195/b048195.txt">Table of n, a(n) for n = 1..10000</a>

%F A034444(A001405(k)) < A048105(A001405(k)).

%e k = 58: binomial(58,29) has 20480 divisors, 8192 unitary ones and 12288 non-unitary ones, and 8192 < 12288.

%t q[n_] := Module[{e = FactorInteger[Binomial[n, Floor[n/2]]][[;; , 2]]}, Times @@ (e + 1) > 2^(Length[e] + 1)]; Select[Range[120], q] (* _Amiram Eldar_, Oct 05 2024 *)

%o (PARI) nbud(n) = 1<<omega(n); \\ from A034444

%o isok(n) = my(b=binomial(n, n\2)); numdiv(b) > 2*nbud(b); \\ _Michel Marcus_, Mar 15 2018

%Y Cf. A001405, A034444, A048105, A048111.

%K nonn

%O 1,1

%A _Labos Elemer_

%E More terms from _Michel Marcus_, Mar 15 2018