A327730 a(n) = A060594(2n).

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

Offset: 1

Author

Jianing Song, Sep 23 2019

Keywords

• nonn
• mult

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.

Examples

			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)).
		

Crossrefs

Cf. A060594, A001221, A001620.

Programs

Mathematica

f[p_, e_] := 2; f[2, e_] := If[e == 1, 2, 4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 31 2022 *)

PARI

a(n) = 2^#znstar(2*n)[2]

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.

From Amiram Eldar, Dec 31 2022: (Start)

Dirichlet g.f.: (zeta(s)^2/zeta(2*s))*((2+2^s+4^s)/(2^s+4^s)).

Sum_{k=1..n} a(k) ~ (n*log(n) + (2*gamma - 5*log(2)/12 - 2*zeta'(2)/zeta(2) - 1)*n)*8/Pi^2, where gamma is Euler's constant (A001620). (End)

Extensions

Offset 1 from Sébastien Palcoux, Jun 22 2022