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1, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 2, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 8, 2, 4, 4, 8, 2, 4, 4, 8, 4, 4, 2, 16, 2, 4, 4, 4, 4, 8, 2, 8, 4, 8, 2, 8, 2, 4, 4, 8, 4, 8, 2, 8, 2, 4, 2, 16, 4, 4, 4, 8, 2, 8, 4, 8, 4, 4, 4, 8, 2, 4, 4, 8
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of quadratic number fields Q(sqrt(d)) (including Q itself) that are subfields of the cyclotomic field Q(exp(Pi*i/n)), where i is the imaginary unit. Note that for odd k, Q(exp(2*Pi*i/k)) = Q(exp(2*Pi*i/(2*k))), so we can just consider the case Q(exp(2*Pi*i/(2*k))) for integers k and let n = 2*k.
a(n) = 2 if and only if n = 2 or n = p^e, where p is an odd prime and e >= 1.
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LINKS
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Table of n, a(n) for n=1..100.
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FORMULA
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a(n) = 2*A060594(n) if n is even and not divisible by 8, otherwise A060594(n).
Multiplicative with a(2) = 2 and a(2^e) = 4 for e > 1; a(p^e) = 2 for odd primes p.
a(n) = 2^omega(n) if 4 does not divide n, otherwise 2^(omega(n)+1), omega = A001221.
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EXAMPLE
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List of quadratic number fields (including Q itself) that are subfields of Q(exp(Pi*i/n)):
n = 2 (the quotient field over the Gaussian integers): Q, Q(i);
n = 3 (the quotient field over the Eisenstein integers): Q, Q(sqrt(-3));
n = 4: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2));
n = 5: Q, Q(sqrt(5));
n = 6: Q, Q(sqrt(3)), Q(sqrt(-3)), Q(i);
n = 7: Q, Q(sqrt(-7));
n = 8: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2));
n = 9: Q, Q(sqrt(-3));
n = 10: Q, Q(sqrt(5)), Q(i), Q(sqrt(-5));
n = 11: Q, Q(sqrt(-11));
n = 12: Q, Q(sqrt(2)), Q(sqrt(3)), Q(sqrt(6)), Q(sqrt(-3)), Q(i), Q(sqrt(-2)), Q(sqrt(-6));
n = 13: Q, Q(sqrt(13));
n = 14: Q, Q(sqrt(7)), Q(i), Q(sqrt(-7));
n = 15: Q, Q(sqrt(5)), Q(sqrt(-3)), Q(sqrt(-15));
n = 16: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2)).
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PROG
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(PARI) a(n) = 2^#znstar(2*n)[2]
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CROSSREFS
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Cf. A060594, A001221.
Sequence in context: A040003 A106469 A082508 * A303809 A193562 A249868
Adjacent sequences: A327727 A327728 A327729 * A327731 A327732 A327733
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KEYWORD
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nonn,mult
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AUTHOR
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Jianing Song, Sep 23 2019
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EXTENSIONS
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Offset 1 from Sébastien Palcoux, Jun 22 2022
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STATUS
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approved
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