A236842 - OEIS

%I #38 Mar 10 2014 08:26:57

%S 0,1,2,3,4,6,7,8,9,11,12,13,14,16,18,19,21,22,24,25,26,27,28,31,32,33,

%T 36,37,38,39,41,42,44,47,48,49,50,52,54,55,56,57,59,61,62,63,64,66,67,

%U 72,73,74,75,76,77,78,81,82,84,87,88,91,93,94,96,97,98,99,100,103

%N Numbers that occur as results of remultiplication (GF(2)[X] -> N) of some number; A234742 sorted and duplicates removed.

%C This sequence gives the range of A234742.

%C After 0 and 1 these are numbers n that have such a multiset of prime divisors p, q, ..., w (p * q * ... * w = n, with p, q, ..., w not necessarily distinct) that it can be arranged so that in at least one subset of divisors of n: (p, q, w), (pq, w), (pw, q), (p, qw), (pqw), ..., all divisors (for example, in the second case: pq and w) encode by their binary representations irreducible factors of polynomial ring over GF(2) (i.e., all occur in A014580) and their (ordinary) product is n.

%C Above condition implies that none of the terms of A091209 occur here.

%H Antti Karttunen, <a href="/A236842/b236842.txt">Table of n, a(n) for n = 1..13487</a>

%F Use the characteristic function A236862(n) to determine whether n is a term of this sequence or not.

%F Specifically:

%F All numbers encoding an irreducible polynomial in GF(2)[X] (A014580) occur in this sequence. This means that a prime is in this sequence if and only if it is in A091206.

%F On the other hand, a composite integer n is in this sequence if and only if it is either in A014580 or it has such a proper factor k (1<k<n, k|n) that both k and n/k are members of this sequence.

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A236842 (NONZERO-POS 1 0 A236862))

%o (define A236842 (NONZERO-POS 1 0 A236853))

%Y Complement: A236844. A236860 is a subsequence.

%Y Positions of nonzero terms in A236853.

%Y Cf. A234742, A236841.

%K nonn

%O 1,3

%A _Antti Karttunen_, Jan 31 2014