A384342 Largest minimum height of the irreducible factors of a degree-n polynomial of height 1.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Mike Speciner, May 26 2025

Keywords

• nonn
• hard
• more

Examples

			For n < 12, every height 1 degree n polynomial has a height 1 irreducible factor, so a(n) = 1.
For n = 12, x^12-x^11-x^9-x^8+x^6-x^4+x^3+x+1 = (x^6-2x^5+x^4-x^2+x-1)(x^6+x^5+x^4-x^2-2x-1) is the product of two irreducible polynomials of height 2, so a(12) >= 2; and every degree 12 height 1 polynomial has an irreducible factor of height at most 2, so a(12) = 2.
		

Crossrefs

Cf. A363959 gives max height of max-height irreducible factor, whereas this sequence gives max height of min-height irreducible factor.

Programs

Python

from msmath.poly import polynomial as poly
def height(p) :
  """find the height, i.e. max abs coeff, of poly p"""
  return max(map(abs, p));
def height1(n) :
  """generate all height 1 polys of degree n"""
  for a in range(3**n) :
    p = [1];
    for i in range(n) :
      a, r = divmod(a, 3);
      p.append(r-1);
    yield poly(*p);
def a(n) :
  """Return max min height of the irreducible factors of a degree n height 1 poly"""
  highest = 0;
  for p in height1(n) :
    f = p.factor();
    h = min(map(height, f));
    if highest < h:
      highest = h;
  return highest;