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

%I #12 May 28 2019 12:20:57

%S 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,

%T 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,

%U 3,1,1,2,2,2,2,2,2,1,2,1,2,2,2,2,1,1,3

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

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

%H Giovanni Resta, <a href="/A308176/b308176.txt">Table of n, a(n) for n = 1..10000</a>

%e The first few factorization are:

%e n, p, factors

%e 1, 2, x^3+x+1

%e 2, 3, x^3+2*x+2

%e 3, 5, (x+3)*(x^2+2*x+3)

%e 4, 7, (x+2)*(x^2+5*x+3)

%e 5, 11, (x+5)*(x^2+6*x+2)

%e 6, 13, x^3+12*x+12

%e 7, 17, (x^2+5*x+7)*(x+12)

%e 8, 19, (x^2+6*x+16)*(x+13)

%e 9, 23, (x+20)*(x+13)^2

%e 10, 29, x^3+28*x+28

%e 11, 31, x^3+30*x+30

%e 12, 37, (x^2+13*x+20)*(x+24)

%e ...

%p p:=x^3-x-1;

%p f:=n->Factor(p) mod ithprime(n);

%p for n from 1 to 20 do lprint(n,ithprime(n),f(n)); od:

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

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

%K nonn

%O 1,3

%A _N. J. A. Sloane_, May 26 2019

%E More terms from _Giovanni Resta_, May 28 2019