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A235032
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Numbers which are factored to the same set of primes in Z as to the binary codes of irreducible polynomials in GF(2)[X].
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11
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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, 48, 52, 56, 59, 61, 62, 64, 67, 73, 74, 76, 82, 88, 94, 96, 97, 103, 104, 109, 111, 112, 118, 122, 123, 124, 128, 131, 134, 137, 146, 148, 152, 157, 164, 167, 176, 188
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OFFSET
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1,3
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COMMENTS
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This is a subsequence of the sequence which gives all such n that A001222(n) = A091222(n).
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LINKS
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EXAMPLE
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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].
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).
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).
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'.
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PROG
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(define A235032 (MATCHING-POS 1 0 (lambda (n) (or (zero? n) (equal? (ifactor n) (GF2Xfactor n))))))
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CROSSREFS
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A235036 and A235039 give composite and odd composite (after 1) terms occurring in this sequence.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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