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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).

%I A358398 #41 Dec 21 2022 21:00:02
%S A358398 15,53,117,215,329,493,657,877,1103,1383,1643,2017,2325,2721,3131,
%T A358398 3601,4009,4575,5031,5647,6221,6849,7409,8211,8849,9593,10335,11199,
%U A358398 11899,12915,13671,14655,15559,16535,17473,18711,19619,20711,21787,23099,24095,25507,26571,27931,29259
%N 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).
%H A358398 Artūras Dubickas, <a href="https://doi.org/10.1007/s00229-014-0657-y">On the number of reducible polynomials of bounded naive height</a>, manuscripta math. 144, 439-456 (2014).
%H A358398 Phyllis Lefton, <a href="https://doi.org/10.1090/S0002-9904-1976-14151-1">On the Galois groups of cubics and trinomials</a>, Bull. Amer. Math. Soc., vol. 82 (1976), pp. 754-756.
%H A358398 Phyllis Lefton, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3533.pdf">On the Galois groups of cubics and trinomials</a>, Acta Arithmetica (1979) Volume: 35, Issue: 3, page 239-246.
%F A358398 Dubickas (2014) shows that a(n) ~ 2(1+(2/3)Pi^2)n^2 = 15.1598... n^2 for large n.
%o A358398 (PARI)
%o A358398 { a(n) =
%o A358398     my( ct = 0 );
%o A358398     for (c1 = -n, n,
%o A358398     for (c2 = -n, n,
%o A358398     for (c3 = -n, n,
%o A358398         if ( ! polisirreducible( Pol([1,c1,c2,c3]) ), ct += 1 );
%o A358398     ); ); );
%o A358398     return( ct );
%o A358398 }
%o A358398 vector(12, n, a(n) ) \\ _Joerg Arndt_, Dec 12 2022
%Y A358398 Cf. A067274.
%K A358398 nonn
%O A358398 1,1
%A A358398 _Lorenz H. Menke, Jr._, Nov 13 2022
%E A358398 a(26)-a(45) from _Hugo Pfoertner_, Nov 27 2022