A396255 Irregular table read by rows: let p = prime(n) and d be the k-th divisor of p-1. Then T(n,k) is the algebraic degree of tau(chi) over Q, where chi is any Dirichlet character modulo p of order d, and tau is the Gauss sum.

1, 1, 2, 1, 2, 8, 1, 2, 6, 12, 1, 2, 20, 40, 1, 2, 6, 8, 12, 48, 1, 2, 8, 32, 128, 1, 2, 6, 12, 54, 108, 1, 2, 110, 220, 1, 2, 8, 42, 84, 336, 1, 2, 6, 20, 12, 40, 120, 240, 1, 2, 6, 8, 12, 54, 48, 108, 432, 1, 2, 8, 20, 32, 40, 160, 640, 1, 2, 6, 12, 42, 84, 252, 504, 1, 2, 506, 1012

Offset

1,3

Comments

If chi has order d, then Q(tau(chi)) is the compositum of Q(zeta_d) and Q(zeta_p)^G, where G is the unique index-d subgroup of (Z/pZ)*. See the Math Stack Exchange link below.

Links

Jianing Song, Table of n, a(n) for n = 1..10001 (Rows 1..713)

Math Stack Exchange, Let chi be a nontrivial Dirichlet charcater of order d modulo p. Is it true that zeta_d is in Q(tau(chi))?.

Formula

T(n,k) = d*EulerPhi(d), where d is the k-th divisor of prime(n)-1.

Example

The first rows:
  1;
  1, 2;
  1, 2, 8;
  1, 2, 6, 12;
  1, 2, 20, 40;
  1, 2, 6, 8, 12, 48;
  1, 2, 8, 32, 128;
  1, 2, 6, 12, 54, 108;
  1, 2, 110, 220;
  1, 2, 8, 42, 84, 336.
See A396254 for explanations of rows 3 and 4.

Prog

(PARI) row(n) = my(p=prime(n), v=divisors(p-1), len=#v); for(i=1, len, v[i]=v[i]*eulerphi(v[i])); v

Crossrefs

Cf. A008328 (row lengths), A002618 ({d*EulerPhi(d)}), A104039 (rightmost values of rows).

Cf. A396254 (another version).

Sequence in context: A030651 A179946 A198757 * A173755 A140894 A208747

Adjacent sequences: A396251 A396253 A396254 * A396256 A396257 A396258

Keyword

nonn,tabf,easy

Author

Jianing Song, May 20 2026

Status

approved