A262405 Least k such that the k-th cyclotomic polynomial has -n as a coefficient.

4, 1, 105, 385, 1365, 2145, 2805, 3135, 6545, 7917, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 17255, 17255, 17255, 20615, 25935, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 0

Author

Charles R Greathouse IV, Sep 21 2015

Keywords

• nonn

Comments

Suzuki proves that a(n) exists for each n.

Examples

			Phi(105) = x^48 + x^47 + x^46 - x^43 - x^42 - 2x^41 - x^40 - x^39 + x^36 + x^35 + x^34 + x^33 + x^32 + x^31 - x^28 - x^26 - x^24 - x^22 - x^20 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 - x^9 - x^8 - 2x^7 - x^6 - x^5 + x^2 + x + 1, with -2 as the coefficient of x^7 (among others), and this is the least k for which -2 appears, so a(2) = 105.
		

Links

• Jiro Suzuki, On coefficients of cyclotomic polynomials, Proc. Japan Acad. Ser. A Math. Sci. 63:7 (1987), pp. 279-280.
• R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).

Crossrefs

Cf. A013594, A013595, A046887, A262404, A160340, A278567.

Programs

Mathematica

Table[k = 1; While[! MemberQ[CoefficientList[Cyclotomic[k, x], x], -n], k++]; k, {n, 0, 9}] (* Michael De Vlieger, Sep 29 2015 *)

PARI

a(n)=my(k,v);while(!setsearch(Set(Vec(polcyclo(k++))),-n),);k

Extensions

More terms from Seiichi Manyama, Dec 22 2018