cp's OEIS Frontend

A235033 Numbers which are factored to a different set of primes in Z as to the irreducible polynomials in GF(2)[X].

%I A235033 #26 Nov 25 2021 08:50:06
%S A235033 5,9,10,15,17,18,20,21,23,25,27,29,30,33,34,35,36,39,40,42,43,45,46,
%T A235033 49,50,51,53,54,55,57,58,60,63,65,66,68,69,70,71,72,75,77,78,79,80,81,
%U A235033 83,84,85,86,87,89,90,91,92,93,95,98,99,100,101,102,105,106
%N A235033 Numbers which are factored to a different set of primes in Z as to the irreducible polynomials in GF(2)[X].
%C A235033 If a term is included in this sequence, then all its ordinary multiples as well as any "A048720-multiples" are included as well. (Cf. the characteristic function A235046.)
%C A235033 The sequence which gives all such n that A001222(n) differs from A091222(n) is a subsequence of this sequence.
%H A235033 Antti Karttunen, <a href="/A235033/b235033.txt">Table of n, a(n) for n = 1..10000</a>
%H A235033 <a href="/index/Ge#GF2X">Index entries for sequences operating on (or containing) GF(2)[X]-polynomials</a>
%e A235033 5 is included in this sequence, because, although it is prime, its binary representation '101' encodes a polynomial x^2 + 1, which is reducible in polynomial ring GF(2)[X] as (x+1)(x+1), i.e., 5 = A048720(3,3).
%e A235033 9 is included in this sequence, as it factors as 3*3 in Z, the corresponding polynomial (bin.repr. '1001'): x^3 + 1 factors as (x+1)(x^2+x+1), i.e., 9 = A048720(3,7), so even although the number of prime/irreducible factors is the same, the factors themselves (i.e., their binary codes) are not exactly the same, thus 9 is included here.
%e A235033 On the other hand, none of 2, 3, 4, 11 and 111 are included in this sequence because they occur in the complement sequence, A235032 (please see examples there).
%o A235033 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A235033 (define A235033 (MATCHING-POS 1 0 (lambda (n) (not (or (zero? n) (equal? (ifactor n) (GF2Xfactor n)))))))
%Y A235033 Gives the positions of nonzeros in A236380, i.e., n such that A234741(n) <> A234742(n).
%Y A235033 Characteristic function: A235046.
%Y A235033 Complement: A235032. Subsets: A091209, A091214.
%K A235033 nonn
%O A235033 1,1
%A A235033 _Antti Karttunen_, Jan 02 2014