A262404 Least k such that the k-th cyclotomic polynomial has n as a coefficient.

4, 1, 165, 595, 1785, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 20615, 17255, 20615, 20615, 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
• look

Comments

Suzuki proves that a(n) exists for each n. Vaughan proves that there are infinitely many k with a(n) = k and n > exp(exp(log 2 * log k/log log k)).

Examples

			Phi(165) = x^80 + x^79 + x^78 - x^75 - x^74 - x^73 - x^69 - x^68 - x^67 + x^65 + 2x^64 + 2x^63 + x^62 - x^60 - x^59 - x^58 - x^54 - x^53 - x^52 + x^50 + 2x^49 + 2x^48 + 2x^47 + x^46 - x^44 - x^43 - x^42 - x^41 - x^40 - x^39 - x^38 - x^37 - x^36 + x^34 + 2x^33 + 2x^32 + 2x^31 + x^30 - x^28 - x^27 - x^26 - x^22 - x^21 - x^20 + x^18 + 2x^17 + 2x^16 + x^15 - x^13 - x^12 - x^11 - x^7 - x^6 - x^5 + x^2 + x + 1, with 2 as the coefficient of x^16 (among others), and this is the least k for which 2 appears, so a(2) = 165.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1000
• 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, A262405, A160340, A278567.

Programs

Maple

N:= 40: count:= 0: A:= Array(0..N): A[0]:= 4:
for k from 1 while count < N do
  S:= select(t -> t::posint and t <= N and A[t] = 0, {coeffs(numtheory:-cyclotomic(k,x),x)}):
  if S <> {} then
    A[convert(S,list)]:= k;
    count:= count + nops(S);
  fi
od:
convert(A,list); # Robert Israel, Dec 23 2018

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

Corrected a(22); more terms from Seiichi Manyama, Dec 22 2018