login
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).
2
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
1,1
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.
FORMULA
Dubickas (2014) shows that a(n) ~ 2(1+(2/3)Pi^2)n^2 = 15.1598... n^2 for large n.
PROG
(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
CROSSREFS
Cf. A067274.
Sequence in context: A333617 A340868 A194454 * A372928 A219384 A198955
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(26)-a(45) from Hugo Pfoertner, Nov 27 2022
STATUS
approved