cp's OEIS Frontend

A279367 Numbers n for which the number of distinct prime divisors of binomial(n,k) has a "local maximum" (see the unusual meaning given in comment) at k = floor(n/2).

%I A279367 #27 Jan 15 2017 13:23:02
%S A279367 1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,30,31,
%T A279367 32,33,35,36,37,39,40,41,42,43,47,48,49,50,55,56,57,58,59,63,64,65,66,
%U A279367 67,68,71,72,73,75,76,80,83,84,85,86,89,90,91,96,97,98,99,100,107,108,109,119
%N A279367 Numbers n for which the number of distinct prime divisors of binomial(n,k) has a "local maximum" (see the unusual meaning given in comment) at k = floor(n/2).
%C A279367 The expression "local maximum" is understood here in a broad sense (as it were considering a flat-topped hill to be the same as a pointed hill): it is assumed that there is a local maximum a(n1) = a1 at n=n1 if there exists a neighborhood of n1 where the sequence is of the form {a0, a1, ..., a1, a2} with a0<a1 and a2<a1. Of course, in the case {a0,a1,a2}, where a1 is not repeated, this coincides with the usual definition of a local maximum in the strict sense.
%C A279367 This sequence is different from A020731, which concerns a global maximum.
%C A279367 Sometimes a local maximum is not the global maximum: for instance, with n = 59, the global maximum omega(binomial(59,22)) = 13 is obtained at k = 22, but there is a local maximum 12 at k = floor(59/2) = 29; this is the first occurrence absent from A020731, the next ones being 86, 91, 121, 123, 169, ... (see the link).
%H A279367 Jean-François Alcover, <a href="/A279367/a279367.pdf">Plot of omega(binomial(59,k))</a>
%t A279367 Select[Range[120], MatchQ[PrimeNu[Binomial[#, Range[Floor[#/2], #]]], {(x_) .., y_, ___} /; x > y] || # == 1&]
%Y A279367 Cf. A001221, A019491, A020731.
%K A279367 nonn
%O A279367 1,2
%A A279367 _Jean-François Alcover_, Dec 10 2016