cp's OEIS Frontend

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

%I A384342 #11 Jul 23 2025 16:09:06
%S A384342 1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2
%N A384342 Largest minimum height of the irreducible factors of a degree-n polynomial of height 1.
%e A384342 For n < 12, every height 1 degree n polynomial has a height 1 irreducible factor, so a(n) = 1.
%e A384342 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.
%o A384342 (Python)
%o A384342 from msmath.poly import polynomial as poly
%o A384342 def height(p) :
%o A384342   """find the height, i.e. max abs coeff, of poly p"""
%o A384342   return max(map(abs, p));
%o A384342 def height1(n) :
%o A384342   """generate all height 1 polys of degree n"""
%o A384342   for a in range(3**n) :
%o A384342     p = [1];
%o A384342     for i in range(n) :
%o A384342       a, r = divmod(a, 3);
%o A384342       p.append(r-1);
%o A384342     yield poly(*p);
%o A384342 def a(n) :
%o A384342   """Return max min height of the irreducible factors of a degree n height 1 poly"""
%o A384342   highest = 0;
%o A384342   for p in height1(n) :
%o A384342     f = p.factor();
%o A384342     h = min(map(height, f));
%o A384342     if highest < h:
%o A384342       highest = h;
%o A384342   return highest;
%Y A384342 Cf. A363959 gives max height of max-height irreducible factor, whereas this sequence gives max height of min-height irreducible factor.
%K A384342 nonn,hard,more
%O A384342 1,12
%A A384342 _Mike Speciner_, May 26 2025