A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 384, 432, 486, 576, 648, 729, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2187, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8748, 9216

Offset

1,1

Comments

Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - Lekraj Beedassy, Sep 10 2004

This is true: see link "Cyclotomic trinomials". - Robert Israel, Jul 14 2015

3-smooth numbers (A003586) which are not powers of 2 (A000079). - Amiram Eldar, Nov 10 2020

These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - Gus Wiseman, Nov 09 2023

References

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Robert Israel, Cyclotomic trinomials

Formula

A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012

a(n) = A033845(n)/2 = 3 * A003586(n). - Robert Israel, Jul 14 2015

Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020

Example

The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
From Gus Wiseman, Nov 09 2023: (Start)
The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
    3: (2)              4: (1,1)
    6: (2,1)            6: (2,1)
    9: (2,2)            9: (2,2)
   12: (2,1,1)         10: (3,1)
   18: (2,2,1)         15: (3,2)
   24: (2,1,1,1)       14: (4,1)
   27: (2,2,2)         25: (3,3)
   36: (2,2,1,1)       21: (4,2)
   48: (2,1,1,1,1)     22: (5,1)
   54: (2,2,2,1)       35: (4,3)
   72: (2,2,1,1,1)     33: (5,2)
   81: (2,2,2,2)       49: (4,4)
   96: (2,1,1,1,1,1)   26: (6,1)
(End)

Maple

with(numtheory): a := []; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m, x))])=3 then a := [op(a), m] fi od; print(a);

Mathematica

max = 5000; Sort[Flatten[Table[2^a 3^b, {a, 0, Floor[Log[2, max]]}, {b, Floor[Log[3, max/2^a]]}]]] (* Alonso del Arte, May 19 2016 *)

Prog

(PARI) isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015
(PARI) list(lim)=my(v=List(), N); for(n=1, logint(lim\1, 3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Aug 07 2015

Crossrefs

Differs at the 18th term from A063996.

Cf. A003586, A033845, A206787.

For primes (A008578) we have conjugates A000079.

For triprimes (A014612) we have conjugates A080193.

A001358 lists semiprimes, squarefree A006881, complement A100959.

Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.

Sequence in context: A052287 A156997 A063996 * A293396 A173195 A354785

Adjacent sequences: A065116 A065117 A065118 * A065120 A065121 A065122

Keyword

nonn

Author

Len Smiley, Nov 12 2001

Extensions

Offset set to 1 and more terms from Michel Marcus, Jul 11 2015

Status

approved