A236842 Numbers that occur as results of remultiplication (GF(2)[X] -> N) of some number; A234742 sorted and duplicates removed.

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, 36, 37, 38, 39, 41, 42, 44, 47, 48, 49, 50, 52, 54, 55, 56, 57, 59, 61, 62, 63, 64, 66, 67, 72, 73, 74, 75, 76, 77, 78, 81, 82, 84, 87, 88, 91, 93, 94, 96, 97, 98, 99, 100, 103

Offset

1,3

Comments

This sequence gives the range of A234742.

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.

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..13487

Formula

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

Specifically:

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.

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.

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A236842 (NONZERO-POS 1 0 A236862))
(define A236842 (NONZERO-POS 1 0 A236853))

Crossrefs

Complement: A236844. A236860 is a subsequence.

Positions of nonzero terms in A236853.

Cf. A234742, A236841.

Sequence in context: A225859 A020655 A188042 * A236860 A031942 A043090

Adjacent sequences: A236839 A236840 A236841 * A236843 A236844 A236845

Keyword

nonn

Author

Antti Karttunen, Jan 31 2014

Status

approved