A278567 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

COMMENTS

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

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.

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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