A395473 - OEIS

A395473

Number of Q(chi) for chi running through the Dirichlet characters modulo n, where Q(chi) is the field generated by the values of chi.

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 1, 4, 4, 3, 2, 4, 2, 4, 3, 2, 4, 4, 2, 6, 3, 4, 2, 6, 2, 4, 2, 4, 2, 2, 2, 4, 4, 4, 4, 4, 3, 4, 2, 3, 4, 2, 2, 8, 4, 2, 4, 4, 2, 4, 4, 2, 4, 4, 2, 9, 6, 4, 3, 4, 4, 4, 2, 4, 6, 2, 2, 4, 4, 4, 2, 6, 4

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OFFSET

1,5

COMMENTS

a(1) = a(2) = 1; for n >= 3, a(n) is the number of even divisors of psi(n) = A002322(n), where A002322 is the reduced totient function. This is because Q(zeta_d) = Q(zeta_(2d)) for odd d.

In other words, number of divisors of psi(n)/2 for n >= 3.

Each odd prime occurs only finitely many times: if psi(n)/2 has exactly p divisors, then psi(n) = 2*P^(p-1) for some prime P. By the nature of the reduced totient function, n has a prime power factor q such that psi(q) = 2*P^(p-1). No prime power with exponent >= 2 can satisfy this equation, so q = 2*P^(p-1) + 1 must be prime. But 2*P^(p-1) + 1 is divisible by 3 for P > 3 since p-1 is even, and so we must have psi(n) = 2^p or 2*3^(p-1). As a result, each odd prime p occurs exactly A395569(2^(p-1)) + A395569(3^(p-1)) times. The p fields Q(chi) are either Q(zeta_2) = Q, Q(zeta_4), ..., Q(zeta_{2^p}) or Q(zeta_2) = Q, Q(zeta_6), ..., Q(zeta_{2*3^(p-1)}).

In particular:

- Since 2^p + 1 is not prime, we have that psi(n) = 2^p if and only if n is 2^(p+2) times a product of distinct Fermat primes 2^2^m + 1 with 2^m <= p-1. If S(p) is the number of such Fermat primes, then the number of solutions to psi(n) = 2^p is 2^S(p).

- psi(n) = 2*3^(p-1) if and only if n = 2^a * 3^b * (a product of distinct primes of the form 2*3^m + 1, 1 <= m <= p-1), where 0 <= a <= 3, and either b = p or 2*3^(p-1) + 1 is a prime factor of n. If T(p) is the number of such primes 2*3^m + 1, then the number of solutions to psi(n) = 2*3^(p-1) is 2^(T(p)+2) if 2*3^(p-1) + 1 is not prime and (p+2) * 2^(T(p)+1) otherwise.

Conjecturally each composite number k occurs infinitely many times. In particular, this would be true if there are infinitely many primes of the form 2^a*3^(b-1)*p1^(e1-1)*...*pr^(er-1), with a*b*e1*...*er = k, b > 1.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Jianing Song, List of Q(chi) for characters chi modulo n <= 2048.

Wikipedia, Dirichlet character.

MATHEMATICA

DivisorSigma[0, Ceiling[CarmichaelLambda[Range[100]]/2]] (* Paolo Xausa, Jun 03 2026 *)

PROG

(PARI) a(n) = if(n<=2, 1, numdiv(A002322(n)/2)) \\ See A002322 for its program

CROSSREFS

Cf. A002322, A066800 (number of divisors of psi), A395569, A395584 (earliest occurrences), A396486.

Cf. A018253 (indices of 1), A395563 (indices of 2), A395464 (indices of 3).

Essentially the same as A191613.