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
Keywords
Examples
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)
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
Crossrefs
Differs at the 18th term from A063996.
Programs
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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);
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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 *) -
PARI
isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015
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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
Formula
A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012
Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020
Extensions
Offset set to 1 and more terms from Michel Marcus, Jul 11 2015
Comments