cp's OEIS Frontend

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

%I A327730 #20 Dec 31 2022 03:39:49
%S A327730 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,
%T A327730 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,
%U A327730 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
%N A327730 a(n) = A060594(2n).
%C A327730 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.
%C A327730 a(n) = 2 if and only if n = 2 or n = p^e, where p is an odd prime and e >= 1.
%F A327730 a(n) = 2*A060594(n) if n is even and not divisible by 8, otherwise A060594(n).
%F A327730 Multiplicative with a(2) = 2 and a(2^e) = 4 for e > 1; a(p^e) = 2 for odd primes p.
%F A327730 a(n) = 2^omega(n) if 4 does not divide n, otherwise 2^(omega(n)+1), omega = A001221.
%F A327730 From _Amiram Eldar_, Dec 31 2022: (Start)
%F A327730 Dirichlet g.f.: (zeta(s)^2/zeta(2*s))*((2+2^s+4^s)/(2^s+4^s)).
%F A327730 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)
%e A327730 List of quadratic number fields (including Q itself) that are subfields of Q(exp(Pi*i/n)):
%e A327730 n = 2 (the quotient field over the Gaussian integers): Q, Q(i);
%e A327730 n = 3 (the quotient field over the Eisenstein integers): Q, Q(sqrt(-3));
%e A327730 n = 4: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2));
%e A327730 n = 5: Q, Q(sqrt(5));
%e A327730 n = 6: Q, Q(sqrt(3)), Q(sqrt(-3)), Q(i);
%e A327730 n = 7: Q, Q(sqrt(-7));
%e A327730 n = 8: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2));
%e A327730 n = 9: Q, Q(sqrt(-3));
%e A327730 n = 10: Q, Q(sqrt(5)), Q(i), Q(sqrt(-5));
%e A327730 n = 11: Q, Q(sqrt(-11));
%e A327730 n = 12: Q, Q(sqrt(2)), Q(sqrt(3)), Q(sqrt(6)), Q(sqrt(-3)), Q(i), Q(sqrt(-2)), Q(sqrt(-6));
%e A327730 n = 13: Q, Q(sqrt(13));
%e A327730 n = 14: Q, Q(sqrt(7)), Q(i), Q(sqrt(-7));
%e A327730 n = 15: Q, Q(sqrt(5)), Q(sqrt(-3)), Q(sqrt(-15));
%e A327730 n = 16: Q, Q(sqrt(2)), Q(i), Q(sqrt(-2)).
%t A327730 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 *)
%o A327730 (PARI) a(n) = 2^#znstar(2*n)[2]
%Y A327730 Cf. A060594, A001221, A001620.
%K A327730 nonn,mult
%O A327730 1,2
%A A327730 _Jianing Song_, Sep 23 2019
%E A327730 Offset 1 from _Sébastien Palcoux_, Jun 22 2022