A013594 - OEIS

%I #34 Sep 07 2023 04:49:58

%S 0,105,385,1365,1785,2805,3135,6545,6545,10465,10465,10465,10465,

%T 10465,11305,11305,11305,11305,11305,11305,11305,15015,11305,17255,

%U 17255,20615,20615,26565,26565,26565,26565,26565,26565,26565,26565,26565

%N Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

%C This sequence is infinite - see the Lang reference.

%C An alternative version would start with 1 rather than 0.

%D Bateman, C. Pomerance and R. C. Vaughan, Colloq. Math. Soc. Janos Bolyai, 34 (1984), 171-202.

%D S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.

%D Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.

%D Maier, Prog. Math. 139 (Birkhaueser) 1996, 633-638.

%H T. D. Noe, <a href="/A013594/b013594.txt">Table of n, a(n) for n = 1..1000</a>

%H P. Erdős and R. C. Vaughan, <a href="http://www.renyi.hu/~p_erdos/1974-03.pdf">Bounds for the r-th coefficients of cyclotomic polynomials</a>, J. London Math. Soc. (2) 8 (1974), 393-400 (MR50 #9835; Zentralblatt 295.10014).

%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

%H H. Maier, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6432.pdf">Cyclotomic polynomials with large coefficients</a>, Acta Arith. 64 (1993), 227-235.

%H H. Maier, <a href="http://dx.doi.org/10.1023/A:1015293917813">Cyclotomic polynomials whose orders contain many prime factors</a>, Period. Math. Hungar. 43 (2001), 155-169.

%H H. L. Montgomery and R. C. Vaughan, <a href="http://dx.doi.org/10.1017/S0017089500006145">The order of magnitude of the mth coefficients of cyclotomic polynomials</a>, Glasgow Math. J. 27 (1985), 143-159.

%H R. C. Vaughan, <a href="http://dx.doi.org/10.1307/mmj/1029001352">Bounds for the coefficients of cyclotomic polynomials</a>, Michigan Math. J. 21 (1974), 289-295 (1975).

%H M. Wallner, <a href="http://dmg.tuwien.ac.at/drmota/Thesis_Wallner.pdf">Lattice Path Combinatorics</a>, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>

%e a(2)=105 because cyclotomic(105) contains "-2" as coefficient, but for n < 105 cyclotomic(n) does not contain 2 or -2.

%e x^105 - 1 = ( - 1 + x)(1 + x + x^2)(1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12)(1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + x^19 - x^23 + x^24)(1 + x + x^2 - x^5 - x^6 - 2x^7 - x^8 - x^9 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 - x^20 - x^22 - x^24 - x^26 - x^28 + x^31 + x^32 + x^33 + x^34 + x^35 + x^36 - x^39 - x^40 - 2x^41 - x^42 - x^43 + x^46 + x^47 + x^48)

%t Table[Position[Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 10000}], j][[1]], {j, 1, 10}] - _Ian Miller_, Feb 25 2008

%o (PARI) nm=6545; m=0; forstep(n=1, nm, 2, if(issquarefree(n), p=polcyclo(n); o=poldegree(p); for(k=0, o, a=abs(polcoeff(p, k)); if(a>m, m=a; print([m, n, factor(n)])))))

%Y Cf. A046887, A013595, A013596, A063696, A063698, A134518, A137979.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Eric W. Weisstein_

%E Further terms from _T. D. Noe_, Oct 29 2007