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%I #21 Feb 05 2014 10:53:48
%S 0,1,2,3,4,6,7,8,11,12,13,14,16,19,22,24,26,28,31,32,37,38,41,44,47,
%T 48,52,56,59,61,62,64,67,73,74,76,82,88,94,96,97,103,104,109,111,112,
%U 118,122,123,124,128,131,134,137,146,148,152,157,164,167,176,188
%N Numbers which are factored to the same set of primes in Z as to the binary codes of irreducible polynomials in GF(2)[X].
%C This is a subsequence of the sequence which gives all such n that A001222(n) = A091222(n).
%H Antti Karttunen, <a href="/A235032/b235032.txt">Table of n, a(n) for n = 1..3048</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on (or containing) GF(2)[X]-polynomials</a>
%e 2, 3 and 11 are included in this sequence, because they occur in A091206. That is, they are all primes, and encode irreducible polynomials in ring GF(2)[X] via their binary representations: For 2, '10' in binary, corresponds to polynomial x, and for 3, '11' in binary, corresponds to polynomial x+1, and for 11, '1011' in binary, corresponds to polynomial x^3+x+1, which are all irreducible in GF(2)[X].
%e 4 is included in this sequence, because it factors as 2*2, but also because the corresponding GF(2)[X] polynomial x^2 factors as x*x (with the polynomial x encoded by the number 2).
%e 5 is NOT included in this sequence, because, although it is prime, the corresponding polynomial (5 in binary is '101'): x^2 + 1 is not irreducible in GF(2)[X], but factors as (x+1)(x+1), i.e., we have 5 = A048720(3,3).
%e 111 is included, as it is a product of two primes, 3*37, and these primes encode via their binary representations, '11' and '100101', two polynomials irreducible in GF(2)[X]: x+1 and x^5 + x^2 + 1, whose product, x^6 + x^5 + x^3 + x^2 + x + 1, is encoded by 111's binary representation, '1101111'.
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A235032 (MATCHING-POS 1 0 (lambda (n) (or (zero? n) (equal? (ifactor n) (GF2Xfactor n))))))
%Y Complement: A235033. Intersection of A235034 & A235035. Union of A091206 & A235036. Subsequence of A235045.
%Y A235036 and A235039 give composite and odd composite (after 1) terms occurring in this sequence.
%Y Gives the positions of zeros in A236380, i.e. such n that A234741(n) = A234742(n).
%Y Cf. also A048720.
%K nonn
%O 1,3
%A _Antti Karttunen_, Jan 02 2014
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