A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 17255, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

This sequence is infinite - see the Lang reference.

An alternative version would start with 1 rather than 0.

Examples

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

References

• Bateman, C. Pomerance and R. C. Vaughan, Colloq. Math. Soc. Janos Bolyai, 34 (1984), 171-202.
• S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.
• Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.
• Maier, Prog. Math. 139 (Birkhaueser) 1996, 633-638.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
• P. Erdős and R. C. Vaughan, Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393-400 (MR50 #9835; Zentralblatt 295.10014).
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227-235.
• H. Maier, Cyclotomic polynomials whose orders contain many prime factors, Period. Math. Hungar. 43 (2001), 155-169.
• H. L. Montgomery and R. C. Vaughan, The order of magnitude of the mth coefficients of cyclotomic polynomials, Glasgow Math. J. 27 (1985), 143-159.
• R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).
• M. Wallner, Lattice Path Combinatorics, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Crossrefs

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

Programs

Mathematica

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 *)

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)])))))

Extensions

More terms from Eric W. Weisstein

Further terms from T. D. Noe, Oct 29 2007