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).

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, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 47, 48, 49, 50, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 71, 72, 73, 75, 76, 80, 83, 84, 85, 86, 89, 90, 91, 96, 97, 98, 99, 100, 107, 108, 109, 119

Offset

1,2

Comments

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.

This sequence is different from A020731, which concerns a global maximum.

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).

Links

Table of n, a(n) for n=1..73.

Jean-François Alcover, Plot of omega(binomial(59,k))

Mathematica

Select[Range[120], MatchQ[PrimeNu[Binomial[#, Range[Floor[#/2], #]]], {(x_) .., y_, ___} /; x > y] || # == 1&]

Crossrefs

Cf. A001221, A019491, A020731.

Sequence in context: A052040 A328781 A069570 * A020731 A229300 A229301

Adjacent sequences: A279364 A279365 A279366 * A279368 A279369 A279370

Keyword

nonn

Author

Jean-François Alcover, Dec 10 2016

Status

approved