A354060 -id:A354060 - cp's OEIS frontend

A354057 Square array read by ascending antidiagonals: T(n,k) is the number of solutions to x^k == 1 (mod n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 1, 1, 4, 1, 4, 1, 4, 1, 2, 1, 2, 1, 1, 1

Offset: 1

Jianing Song, May 16 2022

Row n and Row n' are the same if and only if (Z/nZ)* = (Z/n'Z)*, where (Z/nZ)* is the multiplicative group of integers modulo n.
Given n, T(n,k) only depends on gcd(k,psi(n)). For the truncated version see A354060.
Each column is multiplicative.

			  n/k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
   1   1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
   2   1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
   3   1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2
   4   1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2
   5   1  2  1  4  1  2  1  4  1  2  1  4  1  2  1  4  1  2  1  4
   6   1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2
   7   1  2  3  2  1  6  1  2  3  2  1  6  1  2  3  2  1  6  1  2
   8   1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4
   9   1  2  3  2  1  6  1  2  3  2  1  6  1  2  3  2  1  6  1  2
  10   1  2  1  4  1  2  1  4  1  2  1  4  1  2  1  4  1  2  1  4
  11   1  2  1  2  5  2  1  2  1 10  1  2  1  2  5  2  1  2  1 10
  12   1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4
  13   1  2  3  4  1  6  1  4  3  2  1 12  1  2  3  4  1  6  1  4
  14   1  2  3  2  1  6  1  2  3  2  1  6  1  2  3  2  1  6  1  2
  15   1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8
  16   1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8
  17   1  2  1  4  1  2  1  8  1  2  1  4  1  2  1 16  1  2  1  4
  18   1  2  3  2  1  6  1  2  3  2  1  6  1  2  3  2  1  6  1  2
  19   1  2  3  2  1  6  1  2  9  2  1  6  1  2  3  2  1 18  1  2
  20   1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8  1  4  1  8

Jianing Song, Table of n, a(n) for n = 1..5050 (the first 100 ascending diagonals)

k-th column: A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).
Applying Moebius transform to the rows gives A354059.
Applying Moebius transform to the columns gives A354058.
Cf. A327924.

T(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]))

If (Z/nZ)* = C_{k_1} X C_{k_2} X ... X C_{k_r}, then T(n,k) = Product_{i=1..r} gcd(k,k_r).
T(p^e,k) = gcd((p-1)*p^(e-1),k) for odd primes p. T(2,k) = 1, T(2^e,k) = 2*gcd(2^(e-2),k) if k is even and 1 if k is odd.
A327924(n,k) = Sum_{q|n} T(n,k) * (Sum_{s|n/q} mu(s)/phi(s*q)).

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.

Original entry on oeis.org

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

Jianing Song, May 16 2022

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).

			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;
...

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

Cf. A354058, A002322, A007431, A354060.

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

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))

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

cp's OEIS Frontend

A354057 Square array read by ascending antidiagonals: T(n,k) is the number of solutions to x^k == 1 (mod n).

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