A334006 Triangle read by rows: T(n,k) = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for nonnegative k < n, n >= 1.

1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 1, 3, 1, 1, 4, 1, 5, 1, 5, 1, 5, 1, 9, 1, 3, 1, 3, 1, 7, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 6, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 13, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 3, 1, 3

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 12 2020

Keywords

• nonn
• tabl

Comments

If the sum of proper divisors of q in row q <= q, then q are 1, 2, 3, 4, 5, 8, 16, 17, 32, 64, 128, 256, 257, ...(union of Fermat primes and powers of 2).

Examples

			Triangle T(n,k) begins:
  n\k| 0   1  2  3  4   5  6  7  8   9 10 11 12  13 14 15 16
  ---+------------------------------------------------------
   1 | 1;
   2 | 1,  1;
   3 | 1,  3, 1;
   4 | 1,  2, 1, 3;
   5 | 1,  5, 1, 1, 1;
   6 | 1,  3, 1, 3, 1,  3;
   7 | 1,  7, 1, 3, 1,  3, 1;
   8 | 1,  4, 1, 5, 1,  5, 1, 5;
   9 | 1,  9, 1, 3, 1,  3, 1, 7, 1;
  10 | 1,  5, 1, 1, 1,  5, 1, 1, 1,  5;
  11 | 1, 11, 1, 3, 1,  3, 1, 3, 1,  3, 1;
  12 | 1,  6, 1, 9, 1,  9, 1, 9, 1,  9, 1, 9;
  13 | 1, 13, 1, 1, 1,  5, 1, 1, 1,  5, 1, 1, 1;
  14 | 1,  7, 1, 3, 1,  3, 1, 7, 1,  3, 1, 3, 1, 7;
  15 | 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1;
  16 | 1,  8, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5;
  17 | 1, 17, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1;
  ...
For (n, k) = (7, 3), there are three nonnegative values of m < n such that m^3 == m (mod 7) (namely 0, 1, and 6) and one nonnegative value of m < n such that -m^3 == m (mod 7) (namely 0), so T(7,3) = 3/1 = 3.
		

Crossrefs

Cf. A000012, A000079, A002322, A010684, A019434, A023506, A026741, A182816, A333570, A337454, A337632, A337633, A337820, A337910.

Programs

Magma

[[#[m: m in [0..n-1] | m^k mod n eq m]/#[m: m in [0..n-1] | -m^k mod n eq m]: k in [0..n-1]]: n in [1..17]];

PARI

T(n, k) = sum(m=0, n-1, Mod(m, n)^k == m)/sum(m=0, n-1, -Mod(m, n)^k == m);
matrix(7, 7, n, k, k--; if (k>=n, 0, T(n,k))) \\ to see the triangle \\ Michel Marcus, Apr 17 2020

Extensions

Name corrected by Peter Kagey, Sep 12 2020