cp's OEIS Frontend

A236841 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number; A234741 sorted and duplicates removed.

%I A236841 #19 Apr 27 2019 21:08:18
%S A236841 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,
%T A236841 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U A236841 51,52,53,54,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N A236841 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number; A234741 sorted and duplicates removed.
%C A236841 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.
%C A236841 Above condition implies that none of the terms of A091214 occur here.
%H A236841 Antti Karttunen, <a href="/A236841/b236841.txt">Table of n, a(n) for n = 1..11200</a>
%F A236841 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.
%e A236841 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,)
%e A236841 25 is not a term because it is an irreducible in GF(2)[X], but not a prime in N.
%e A236841 43 = 3 x 25 is a term because 43 itself is a prime in N.
%e A236841 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.
%e A236841 1951 = 25 x 87 is a member, as although both 25 and 87 are in A091214, 1951 is itself a prime in N.
%o A236841 (Scheme, two different implementations, using _Antti Karttunen_'s IntSeq-library)
%o A236841 (define A236841 (NONZERO-POS 1 0 A236833))
%o A236841 (define A236841 (NONZERO-POS 1 0 A236861))
%Y A236841 Positions of nonzero terms in A236833.
%Y A236841 Complement of A236834.
%Y A236841 Characteristic function: A236861.
%Y A236841 A subsequence: A236839.
%Y A236841 Cf. A236842, A234741, A236833.
%K A236841 nonn
%O A236841 1,3
%A A236841 _Antti Karttunen_, Jan 31 2014