A278567 Maximal coefficient (in absolute value) of cyclotomic polynomial C(N,x), where N = n-th number which is a product of exactly three distinct primes = A007304(n).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 1

Author

N. J. A. Sloane, Nov 26 2016

Keywords

• nonn

Comments

E. Lehmer (1936) shows that this sequence is unbounded.

Examples

			The first 2 occurs in the famous C(105,x), which is x^48+x^47+x^46-x^43-x^42-2*x^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^8-2*x^7-x^6-x^5+x^2+x+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 42 (1936), 389-392.

Crossrefs

See A278571 for smallest m such that a(m) = n.

Cf. A007304, A013595, A160340, A262404, A262405.

See A278570 for another version.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, b(n-1)) while
      bigomega(k)<>3 or nops(factorset(k))<>3 do od; k
    end:
a:= n-> max(map(abs, [coeffs(cyclotomic(b(n), x))])):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 26 2016

Mathematica

f[n_] := Max[ Abs[ CoefficientList[ Cyclotomic[n, x], x]]]; t = Take[ Sort@ Flatten@ Table[Prime@i Prime@j Prime@k, {i, 3, 35}, {j, 2, i -1}, {k, j -1}], 105]; f@# & /@ t (* Robert G. Wilson v, Dec 09 2016 *)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, cyclotomic_poly
def A278567(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return max(int(abs(x[1][0][0])) for x in cyclotomic_poly(bisection(f)).as_terms()[0]) # Chai Wah Wu, Aug 31 2024