A358398 a(n) is the number of reducible monic cubic polynomials x^3 + r*x^2 + s*x + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

15, 53, 117, 215, 329, 493, 657, 877, 1103, 1383, 1643, 2017, 2325, 2721, 3131, 3601, 4009, 4575, 5031, 5647, 6221, 6849, 7409, 8211, 8849, 9593, 10335, 11199, 11899, 12915, 13671, 14655, 15559, 16535, 17473, 18711, 19619, 20711, 21787, 23099, 24095, 25507, 26571, 27931, 29259

Offset: 1

Author

Lorenz H. Menke, Jr., Nov 13 2022

Keywords

• nonn

Links

• Artūras Dubickas, On the number of reducible polynomials of bounded naive height, manuscripta math. 144, 439-456 (2014).
• Phyllis Lefton, On the Galois groups of cubics and trinomials, Bull. Amer. Math. Soc., vol. 82 (1976), pp. 754-756.
• Phyllis Lefton, On the Galois groups of cubics and trinomials, Acta Arithmetica (1979) Volume: 35, Issue: 3, page 239-246.

Crossrefs

Cf. A067274.

Programs

PARI

{ a(n) =
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3]) ), ct += 1 );
    ); ); );
    return( ct );
}
vector(12, n, a(n) ) \\ Joerg Arndt, Dec 12 2022

Formula

Dubickas (2014) shows that a(n) ~ 2(1+(2/3)Pi^2)n^2 = 15.1598... n^2 for large n.

Extensions

a(26)-a(45) from Hugo Pfoertner, Nov 27 2022