A238098 Number of cubic polynomials with coefficients from {1..n} for which all three roots are integers.

0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 19, 21, 23, 25, 27, 30, 34, 36, 39, 44, 46, 49, 54, 57, 60, 64, 67, 72, 76, 79, 85, 91, 92, 95, 100, 106, 109, 115, 117, 122, 129, 132, 136, 147, 150, 154, 159, 163, 166, 174, 180, 187, 191, 194, 199, 210, 211, 216

Offset: 1

Author

N. J. A. Sloane, Feb 22 2014

Keywords

• nonn

Comments

A generalization of A006218 and A238096.

Links

• Dorin Andrica and Eugen J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, Vol. 22(1),2014, 13-23.

Crossrefs

Cf. A006218, A238096, A238097.

Programs

PARI

f(n) = if( n<1, 0, sum(a1=1, n, sum(a2=1, n, sum(a3=1, n, vecmax([a1, a2, a3]) == n && vecsum( factor( Pol([1, a1, a2, a3]))[, 2]) == 3)))); \\ A238097
a(n) = sum(k=1, n, (n\k)*f(k));
lista(nn) = my(v = vector(nn, k, f(k))); vector(nn, i, sum(k=1, i, (i\k)*v[k])); \\ Michel Marcus, Sep 28 2023

Formula

a(n) = Sum_{k=1..n} floor(n/k)*A238097(k).