A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.

1, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 2, 1, 0, 5, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 1, 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15

Offset

1,10

Comments

Given n, T(n,k) only depends on gcd(k,psi(n)).

The n-th row contains entirely 0's if and only if n == 2 (mod 4).

If n !== 2 (mod 4), T(n,psi(n)) > T(n,k) for 1 <= k < psi(n).

Links

Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows)

Formula

For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.

T(n,psi(n)) = A007431(n). - Jianing Song, May 24 2022

Example

Table starts
n = 1: 1;
n = 2: 0;
n = 3: 0, 1;
n = 4: 0, 1;
n = 5: 0, 1, 0, 3;
n = 6: 0, 0;
n = 7: 0, 1, 2, 1, 0, 5;
n = 8: 0, 2;
n = 9: 0, 0, 2, 0, 0, 4;
n = 10: 0, 0, 0, 0;
n = 11: 0, 1, 0, 1, 4, 1, 0, 1, 0, 9;
n = 12: 0, 1;
n = 13: 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11;
n = 14: 0, 0, 0, 0, 0, 0;
n = 15: 0, 1, 0, 3;
n = 16: 0, 0, 0, 4;
n = 17: 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15;
n = 18: 0, 0, 0, 0, 0, 0;
n = 19: 0, 1, 2, 1, 0, 5, 0, 1, 8, 1, 0, 5, 0, 1, 2, 1, 0, 17;
n = 20: 0, 1, 0, 3;
...

Prog

(PARI) b(n, k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n, k) = sumdiv(n, d, moebius(n/d)*b(d, k))

Crossrefs

Cf. A354058, A002322, A007431, A354060.

A354257 gives the smallest index for the nonzero terms in each row.

Sequence in context: A052998 A290174 A344937 * A318921 A114516 A218788

Adjacent sequences: A354058 A354059 A354060 * A354062 A354063 A354064

Keyword

nonn,tabf

Author

Jianing Song, May 16 2022

Status

approved