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%I #58 Nov 12 2023 21:51:04
%S 3,6,9,12,18,24,27,36,48,54,72,81,96,108,144,162,192,216,243,288,324,
%T 384,432,486,576,648,729,768,864,972,1152,1296,1458,1536,1728,1944,
%U 2187,2304,2592,2916,3072,3456,3888,4374,4608,5184,5832,6144,6561,6912,7776,8748,9216
%N Numbers k such that the k-th cyclotomic polynomial is a trinomial.
%C Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - _Lekraj Beedassy_, Sep 10 2004
%C This is true: see link "Cyclotomic trinomials". - _Robert Israel_, Jul 14 2015
%C 3-smooth numbers (A003586) which are not powers of 2 (A000079). - _Amiram Eldar_, Nov 10 2020
%C These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - _Gus Wiseman_, Nov 09 2023
%D Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.
%H Robert Israel, <a href="/A065119/b065119.txt">Table of n, a(n) for n = 1..10000</a>
%H Robert Israel, <a href="/A065119/a065119.pdf">Cyclotomic trinomials</a>
%F A206787(a(n)) = 4. - _Reinhard Zumkeller_, Feb 12 2012
%F a(n) = A033845(n)/2 = 3 * A003586(n). - _Robert Israel_, Jul 14 2015
%F Sum_{n>=1} 1/a(n) = 1. - _Amiram Eldar_, Nov 10 2020
%e The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
%e From _Gus Wiseman_, Nov 09 2023: (Start)
%e The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
%e 3: (2) 4: (1,1)
%e 6: (2,1) 6: (2,1)
%e 9: (2,2) 9: (2,2)
%e 12: (2,1,1) 10: (3,1)
%e 18: (2,2,1) 15: (3,2)
%e 24: (2,1,1,1) 14: (4,1)
%e 27: (2,2,2) 25: (3,3)
%e 36: (2,2,1,1) 21: (4,2)
%e 48: (2,1,1,1,1) 22: (5,1)
%e 54: (2,2,2,1) 35: (4,3)
%e 72: (2,2,1,1,1) 33: (5,2)
%e 81: (2,2,2,2) 49: (4,4)
%e 96: (2,1,1,1,1,1) 26: (6,1)
%e (End)
%p 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);
%t 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 *)
%o (PARI) isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ _Michel Marcus_, Jul 11 2015
%o (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
%Y Differs at the 18th term from A063996.
%Y Cf. A003586, A033845, A206787.
%Y For primes (A008578) we have conjugates A000079.
%Y For triprimes (A014612) we have conjugates A080193.
%Y A001358 lists semiprimes, squarefree A006881, complement A100959.
%Y Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.
%K nonn
%O 1,1
%A _Len Smiley_, Nov 12 2001
%E Offset set to 1 and more terms from _Michel Marcus_, Jul 11 2015
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