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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 0,2 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 STATUS approved

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)