A396486 Irregular triangle read by rows: T(1,1) = T(2,1) = 1; for n >= 3, the n-th row lists the number of Dirichet characters chi modulo n such that Q(chi) = Q(zeta_(2d)) for each divisor d of psi(n)/2, where Q(chi) is the field generated by the values of chi, and psi = A002322 is the reduced totient function.

1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 2, 8, 4, 2, 2, 4, 4, 2, 4, 4, 4, 4, 4, 2, 2, 4, 8, 2, 4, 2, 4, 12, 4, 4, 4, 8, 2, 8, 2, 20, 8, 2, 2, 8, 8, 2, 2, 4, 4, 2, 4, 12, 4, 8, 2, 2, 12, 12, 4, 4, 2, 4, 8, 16, 4, 4, 8, 4, 16, 2, 2, 4, 8, 4, 4, 8, 8, 4, 8, 2, 2, 4, 4, 12, 12

Offset

1,3

Comments

For n >= 3, let d is the k-th divisor of psi(n)/2. If d is odd, then T(n,k) is the number of elements in (Z/nZ)* having order d or 2*d; otherwise T(n,k) is the number of elements in (Z/nZ)* having order 2*d.

Links

Jianing Song, Table of n, a(n) for n = 1..9887 (rows 1..1500)

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

Wikipedia, Dirichlet character.

Example

Table begins
  1,
  1,
  2,
  2,
  2, 2,
  2,
  2, 4,
  4,
  2, 4,
  2, 2,
  2, 8,
  4,
  2, 2, 4, 4,
  2, 4,
  4, 4,
  4, 4,
  2, 2, 4, 8,
  2, 4,
  2, 4, 12,
  4, 4.

Prog

(PARI) power_solution(invariant_factors, k) = vecprod(apply(x->gcd(x, k), invariant_factors)); \\ number of solutions to x^k = 1 in C_{k_1} X ... C_{k_r}
count_order(invariant_factors, k) = sumdiv(k, d, moebius(k/d)*power_solution(invariant_factors, d)); \\ number of elements of order k
row(n) = if(n<=2, [1], my(invariant_factors=znstar(n)[2], k=lcm(invariant_factors)/2); apply(x->count_order(invariant_factors, 2*x)+if(x%2==1, count_order(invariant_factors, x)), divisors(k)));

Crossrefs

Cf. A002322, A265120.

Cf. A060594 (first column), A395473 (row lengths), A395683 (rightmost elements of rows).

Sequence in context: A036263 A307378 A339049 * A279401 A168514 A326353

Adjacent sequences: A396482 A396483 A396484 * A396487 A396488 A396489

Keyword

nonn,tabf

Author

Jianing Song, May 28 2026

Status

approved