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A327816
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Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(8) (counted with multiplicity).
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2
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1, 1, 1, 2, 1, 1, 6, 4, 3, 1, 1, 2, 3, 6, 2, 8, 2, 3, 3, 2, 6, 1, 2, 4, 1, 3, 3, 12, 1, 2, 6, 16, 2, 2, 6, 6, 3, 3, 6, 4, 2, 6, 3, 2, 6, 2, 2, 8, 6, 1, 4, 6, 1, 3, 2, 24, 6, 1, 1, 4, 3, 6, 18, 32, 12, 2, 3, 4, 2, 6, 2, 12, 24, 3, 2, 6, 6, 6, 6, 8, 3, 2, 1, 12, 8, 3, 2, 4, 8, 6
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OFFSET
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1,4
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LINKS
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FORMULA
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Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(8,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
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EXAMPLE
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Let GF(8) = GF(2)[y]/(y^3+y+1). Factorizations of the n-th cyclotomic polynomial over GF(8) for n <= 10:
n = 1: x + 1;
n = 2: x + 1;
n = 3: x^2 + x + 1;
n = 4: (x + 1)^2;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: x^2 + x + 1;
n = 7: (x + y)*(x + (y+1))*(x + y^2)*(x + (y^2+1))*(x + (y^2+y))*(x + (y^2+y+1));
n = 8: (x + 1)^4;
n = 9: (x^2 + y*x + 1)*(x^2 + (y+1)*x + 1)*(x^2 + y^2*x + 1);
n = 10: x^4 + x^3 + x^2 + x + 1.
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MATHEMATICA
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a[n_] := EulerPhi[n] / MultiplicativeOrder[8, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
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PROG
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(PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(8, s))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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