OFFSET
1,3
COMMENTS
The range of A234741: numbers n which encode by their binary representation a polynomial in GF(2)[X] whose multiset of irreducible polynomial factors P, Q, ..., W (where n = P x Q x ... x W, where P, Q, ..., W are irreducible polynomials encoded by A014580, and are not necessarily distinct, and x stands for carryless multiplication of such polynomials: A048720) can be grouped to at least one such multiset (P x Q, W), (P x W, Q), (P, Q x W), (P x Q x W), etc., in such a way that all its members are primes.
Above condition implies that none of the terms of A091214 occur here.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..11200
FORMULA
Use the characteristic function A236861(n) to determine whether n is a term of this sequence or not. Specifically, all primes occur in this sequence. A composite number n occurs only if there exists at least one such pair of k, m < n that n = A048720(k,m) and k and m both occur here. This implies that none of the terms of A091214 are present.
EXAMPLE
17 is a term because it factors as 3 x 3 x 3 x 3 in GF(2)[X], and these can be grouped as (3x3x3x3), (3x3 * 3x3), (3 * 3 * 3x3) and (3 * 3 * 3 * 3) that is, as 17, (5 * 5), (3 * 3 * 5) and (3 * 3 * 3 * 3) which give the four different k, 17, 25, 45 and 81, for which A234741(k) = 17. (Note that A236833(17) = 4. In the grouping (3 * 3x3x3) = (3 * 15) 15 is not a prime, so it is discarded,)
25 is not a term because it is an irreducible in GF(2)[X], but not a prime in N.
43 = 3 x 25 is a term because 43 itself is a prime in N.
125 = 3 x 3 x 25 is a term, because both 3 and (3 x 25) = 43 are primes in N. Their product 3*43 = 129 gives one such k that A234741(k) = 125.
1951 = 25 x 87 is a member, as although both 25 and 87 are in A091214, 1951 is itself a prime in N.
PROG
(Scheme, two different implementations, using Antti Karttunen's IntSeq-library)
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 31 2014
STATUS
approved