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A305421
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GF(2)[X] factorization prime shift towards larger terms.
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10
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1, 3, 7, 5, 21, 9, 11, 15, 49, 63, 13, 27, 19, 29, 107, 17, 273, 83, 25, 65, 69, 23, 121, 45, 31, 53, 151, 39, 35, 189, 37, 51, 251, 819, 173, 245, 41, 43, 233, 195, 47, 207, 93, 57, 997, 139, 55, 119, 127, 33, 1911, 95, 79, 441, 59, 105, 367, 101, 61, 455, 67, 111, 475, 85, 1281, 269, 73, 1365, 81, 503, 457, 287, 87, 123, 1549, 125, 179, 315
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OFFSET
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1,2
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COMMENTS
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Permutation of the odd numbers, A005408.
Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as A014580(i) x A014580(j) x ... x A014580(k), 1 <= i <= j <= ... <= k, a(n) = A014580(1+i) x A014580(1+j) x ... x A014580(1+k).
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LINKS
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FORMULA
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For all n >= 1:
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EXAMPLE
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For n = 12, which by its binary representation '1100' corresponds with (0,1)-polynomial x^3 + x^2, which over GF(2)[X] is factored as (x)(x)(x+1), i.e., 12 = A048720(2,A048720(2,3)) = A048720(A014580(1), A048720(A014580(1),A014580(2))), we then form a(12) as A048720(A014580(2), A048720(A014580(2),A014580(3))) = A048720(3,A048720(3,7)) = 27. Note that x, x+1 and x^2 + x + 1 are the three smallest irreducible (0,1)-polynomials when factored over GF(2)[X], and their binary representations 2, 3 and 7 are the three initial terms of A014580.
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PROG
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(PARI)
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))), x, 2)); for(i=1, #f~, f[i, 1] = Pol(binary(A305420(f[i, 1])))); fromdigits(Vec(factorback(f))%2, 2); };
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CROSSREFS
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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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