A178868 Numbers n such that the trinomial x^n + A x + B has an irreducible cubic as its lowest-degree factor (for some nonzero integers A,B).

6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 22, 24, 52

Offset

1,1

Comments

a(1)=6 because e.g. x^6+16x+16 have factor x^3-2x^2+4

a(2)=7 because e.g. x^7-183352x+814800 have factor x^3+4x^2+22x-300

a(3)=8 because e.g. x^8+7x-4 have factor x^3-x^2+2x-1

a(4)=9 because e.g. x^9+2187x+2916 have factor x^3+3x^2+9x+9

a(5)=10 because e.g. x^10+297x-243 have factor x^3+3x-3 (*Schinzel*)

a(6)=12 because e.g. x^12+576x+368 have factor x^3+2x^2+4x+2 (*Browkin-Schinzel*)

a(7)=13 because e.g. x^13+768x+1024 have factor x^3+2x^2+4x+4 (*Browkin*)

a(8)=14 because x^14+4x+3 have factor x^3-x^2+1 (*Bremner*)

a(9)=15 because x^15-1059328125x+2378362500 have factor x^3+15x-45 (*Browkin*)

a(10)=16 because x^16+3486328125x+9277343750 have factor x^3+5x^2+25x+50 (*Bremner*)

a(11)=17 because x^17+103x+56 have factor x^3-x^2+x+1 (*Bremner*)

a(12)=22 because x^22+376832x-425984 have factor x^3+ 2 x^2-4 (*Browkin*)

a(13)=24 because x^24+14336x+12032 have factor x^3-2x^2+2 (*Browkin-Schinzel*)

a(14)=52 because x^52+2731599200256x+3539053051904 have factor x^3+2x^2+4x+4 (*Browkin*)

References

Schinzel A. 1993. On reducible trinomials. Dissertationes Mathematicae. Warszawa Vol. CCCXXIX, pp.1-83.

Schinzel A. 2000. On reducible trinomials, II. Publicationes Mathematicae. Debrecen. Tomus 56 Fasc.3-4, pp.575-608.

Links

Table of n, a(n) for n=1..14.

Example

a(3)=8 because e.g. x^8+36x-13 has the cubic factor x^3-x^2+3x-1.

Crossrefs

Sequence in context: A269801 A250048 A051053 * A069838 A067901 A115840

Adjacent sequences: A178865 A178866 A178867 * A178869 A178870 A178871

Keyword

more,nonn

Author

Artur Jasinski, Jun 20 2010,Jun 24 2010

Extensions

Definition edited by N. J. A. Sloane, Jun 25 2010

11 inserted Artur Jasinski, Jun 25 2010

16 inserted and 11 deleted Artur Jasinski, Jun 29 2010

Status

approved