A238097 Number of monic cubic polynomials with coefficients from {1..n} and maximum coefficient equal to n, for which all three roots are integers.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 0, 2, 2, 2, 0, 3, 2, 1, 2, 3, 1, 3, 0, 3, 3, 1, 1, 4, 3, 1, 1, 3, 2, 3, 1, 2, 3, 2, 0, 4, 5, 2, 2, 2, 1, 3, 3, 3, 3, 1, 0, 5, 4, 1, 2, 4, 4, 3, 1, 2, 2, 3, 1, 5, 6, 1, 2, 3, 2, 3, 1, 4, 6, 2, 0, 5, 5, 1, 1, 3

Offset

1,11

Links

Zak Seidov, Table of n, a(n) for n = 1..1000

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.

Example

a(11) = 2 with polynomials x^3 + 6*x^2 + 11*x + 6 = (x+1) * (x+2) * (x+3) and x^3 + 7*x^2 + 11*x + 5 = (x+1)^2 * (x+5). - Michael Somos, Feb 23 2014

Mathematica

Table[p = Flatten[Table[{a, b, c, 1}, {a, n}, {b, n}, {c, n}], 2]; cnt = 0; Do[If[Max[p[[i]]] == n, poly = p[[i]].x^Range[0, 3]; r = Rest[FactorList[poly]]; If[Total[Transpose[r][[2]]] == 3 && Union[Coefficient[Transpose[r][[1]], x]] == {1}, Print[{n, r}]; cnt++]], {i, Length[p]}]; cnt, {n, 20}] (* T. D. Noe, Feb 22 2014 *)

Prog

(PARI) {a(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))))}; /* Michael Somos, Feb 23 2014 */

Crossrefs

Cf. A006218, A066955, A238096, A238098.

Sequence in context: A325613 A305054 A375148 * A066955 A089048 A329443

Adjacent sequences: A238094 A238095 A238096 * A238098 A238099 A238100

Keyword

nonn

Author

N. J. A. Sloane, Feb 22 2014

Extensions

Definition corrected by Giovanni Resta, Feb 22 2014

Extended by T. D. Noe, Feb 22 2014

Status

approved