A267509 Integers whose base-10 representation (Bm,...,B1,B0) is such that the polynomial f(x) = B0 + B1*x + ... + Bm*x^m is reducible over the ring of integers, 0 <= Bi <= 9.

4, 6, 8, 9, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 42, 44, 46, 48, 50, 55, 60, 62, 63, 64, 66, 68, 69, 70, 77, 80, 82, 84, 86, 88, 90, 93, 96, 99, 100, 110, 120, 121, 130, 132, 140, 143, 144, 150, 154, 156, 160, 165, 168, 169, 170, 176, 180, 187, 190, 198, 200

Offset: 1

Author

Abdul Gaffar Khan, Jan 16 2016

Keywords

• nonn
• base

Examples

			4 is a term as f(x) = B0 = 4 = 2*2 = g(x)*h(x) with g(x)=h(x)=2 and neither g(x) nor h(x) is a unit in the integer ring. This implies that f(x) is reducible over the ring of integers.
22 is a term as f(x) = B0 + B1*x = 2 + 2*x = 2(1+x) = g(x)*h(x) with g(x)=2 and h(x)=1+x.
110 is a term as f(x) = B0 + B1*x + B2*x^2 = 0 + 1*x + 1*x^2 = x + x^2 = x(1+x) = g(x)*h(x) with g(x)=x and h(x)=1+x.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A121719.

Programs

Mathematica

okQ[n_] := n<10 && CompositeQ[n] || MatchQ[Factor[(id = IntegerDigits[n]). x^Range[Length[id]-1, 0, -1]][[0]], Times|Power]; Select[Range[250], okQ] (* Jean-François Alcover, Feb 01 2016 *)

PARI

isok(n) = {p = Pol(digits(n)); if (poldegree(p) == 0, return ((n!=1) && !isprime(n))); if (! polisirreducible(p), return (1)); f = factor(p); q = prod(k=1, #f~, f[k,1]^f[k,2]); r = p/q; nr = polcoeff(r, 0); if (nr != 1, return (1));} \\ Michel Marcus, Jan 31 2016

PARI

isok(n) = {d = digits(n); p = Pol(d); if (poldegree(p) == 0, return ((n!=1) && !isprime(n))); if (! polisirreducible(p), return (1)); return (gcd(d) != 1);} \\ Michel Marcus, Feb 01 2016