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 A327730 a(n) = A060594(2n). 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS FORMULA 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. EXAMPLE 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)). PROG (PARI) a(n) = 2^#znstar(2*n) CROSSREFS Cf. A060594, A001221. Sequence in context: A040003 A106469 A082508 * A303809 A193562 A249868 Adjacent sequences:  A327727 A327728 A327729 * A327731 A327732 A327733 KEYWORD nonn,mult AUTHOR Jianing Song, Sep 23 2019 EXTENSIONS Offset 1 from Sébastien Palcoux, Jun 22 2022 STATUS approved

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Last modified August 9 21:08 EDT 2022. Contains 356026 sequences. (Running on oeis4.)