A284367 Square array T(n,k) = number of separable polynomials of degree <= k in Z/n[x], n>=1, k>=1, read by antidiagonals.

1, 1, 3, 1, 5, 8, 1, 9, 20, 12, 1, 17, 56, 40, 24, 1, 33, 164, 144, 104, 24, 1, 65, 488, 544, 504, 100, 48, 1, 129, 1460, 2112, 2504, 504, 300, 48, 1, 257, 4376, 8320, 12504, 2788, 2064, 320, 72, 1, 513, 13124, 33024, 62504, 16104, 14412, 2304, 540, 72

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Leonard Carlitz, The arithmetic of polynomials in a Galois field, Amer. J. Math., 54(1):39-50, January 1932.

Jason K. C. Polak, Counting Separable Polynomials in Z/n[x], arXiv:1703.07064 [math.RA], 2016.

Example

   1,   1,   1,    1,     1, ...
   3,   5,   9,   17,    33, ...
   8,  20,  56,  164,   488, ...
  12,  40, 144,  544,  2112, ...
  24, 104, 504, 2504, 12504, ...
  24, 100, 504, 2788, 16104, ...

Mathematica

T[n_, k_] := With[{f = FactorInteger[n][[All, 1]]}, EulerPhi[n] n^k Times @@ (1 + 1/f^k)]; T[1, _] = 1;
Table[T[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 28 2018, from PARI *)

Prog

(PARI) T(n, k) = my(f=factor(n)); eulerphi(n)*n^k*prod(i=1, #f~, 1+1/f[i, 1]^k);

Crossrefs

Cf. A007434 (1st column).

Sequence in context: A116647 A063858 A209831 * A280328 A280384 A124420

Adjacent sequences: A284364 A284365 A284366 * A284368 A284369 A284370

Keyword

nonn,tabl

Author

Michel Marcus, Mar 25 2017

Status

approved