A308176 Number of factors when x^3-x-1 is factorized mod the n-th prime.

1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 2, 2, 1, 1, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 3

Offset

1,3

References

M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge, 1989; see page 131.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Example

The first few factorization are:
n, p, factors
1, 2, x^3+x+1
2, 3, x^3+2*x+2
3, 5, (x+3)*(x^2+2*x+3)
4, 7, (x+2)*(x^2+5*x+3)
5, 11, (x+5)*(x^2+6*x+2)
6, 13, x^3+12*x+12
7, 17, (x^2+5*x+7)*(x+12)
8, 19, (x^2+6*x+16)*(x+13)
9, 23, (x+20)*(x+13)^2
10, 29, x^3+28*x+28
11, 31, x^3+30*x+30
12, 37, (x^2+13*x+20)*(x+24)
...

Maple

p:=x^3-x-1;
f:=n->Factor(p) mod ithprime(n);
for n from 1 to 20 do lprint(n, ithprime(n), f(n)); od:

Mathematica

a[n_] := Total[Last /@ FactorList[x^3-x-1, Modulus -> Prime[n]]] - 1; Array[a, 100] (* Giovanni Resta, May 28 2019 *)

Prog

(PARI) a(n) = vecsum(factor((x^3-x-1)*Mod(1, prime(n)))[, 2]); \\ Michel Marcus, May 28 2019

Crossrefs

Sequence in context: A116858 A182134 A189684 * A354257 A106493 A309981

Adjacent sequences: A308173 A308174 A308175 * A308177 A308178 A308179

Keyword

nonn

Author

N. J. A. Sloane, May 26 2019

Extensions

More terms from Giovanni Resta, May 28 2019

Status

approved