A107748 Number of monic divisors of x^n - 1 with coefficients in {0,1,-1}.

2, 4, 4, 8, 4, 14, 4, 16, 8, 14, 4, 48, 4, 14, 14, 32, 4, 50, 4, 48, 14, 14, 4, 162, 8, 14, 16, 48, 4, 136, 4, 64, 14, 14, 14, 286, 4, 14, 14, 160, 4, 136, 4, 48, 48, 14, 4, 550, 8, 50, 14, 48, 4, 186, 14, 164, 14, 14, 4, 1124, 4, 14, 48, 128, 14, 136, 4, 48, 14, 136

Offset: 1

Author

W. Edwin Clark, Jun 11 2005

Keywords

• nonn

Comments

Multiply by 2 to get all that have coefficients in {0,1,-1}.

Note that many of these are equal to 2^tau(n), where tau(n) is the number of positive divisors of n = number of irreducible factors of x^n - 1. This is connected with the fact that for small values of n the coefficients of the n-th cyclotomic polynomial belong to {0,1,-1}.

From Robert Israel, Aug 24 2017: (Start)

Each of these polynomials is a product of distinct cyclotomic polynomials C_k(x) for k dividing n.

a(n) <= 2^tau(n).

If n is prime then a(n)=4. (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..719 (n=1..359 from Antti Karttunen)

Crossrefs

Cf. A107067, A067824.

Programs

Maple

f:= proc(n) local t, C, x, S;
  C:= map(m -> numtheory:-cyclotomic(m, x), numtheory:-divisors(n) );
  t:= 0:
  S:= combinat:-subsets(C);
  while not S[finished] do
  if map(abs,{coeffs(expand(convert(S[nextvalue](), `*`)), x)}) = {1} then
    t:= t+1;
  fi
od;
t
end proc:
map(f, [$1..100]); # Robert Israel, Aug 24 2017

PARI

for(n=1, 359, m=0; p=x^n-1; nE=numdiv(n); P=factor(p); E=P[, 2]; P=P[, 1]; forvec(v=vector(nE, i, [0, E[i]]), divp=prod(k=1, nE, P[k]^v[k]); m++; for(j=0, poldegree(divp), divpcof=polcoeff(divp, j); if(divpcof<-1 || divpcof>1, m--; break))); write("b107748.txt", n, " ", m)); \\ Antti Karttunen, Aug 24 2017, after Herman Jamke's PARI-program for A107067