A248015 Positive numbers n such that n^2 + 1 is composite and there are no positive integers c and z such that n = c*z^2 + z + c.

8, 18, 28, 30, 34, 44, 46, 48, 50, 58, 60, 64, 68, 70, 76, 78, 86, 88, 96, 98, 100, 104, 108, 114, 118, 128, 136, 144, 148, 158, 164, 166, 168, 178, 186, 188, 190, 194, 196, 198, 200

Offset: 1

Author

Matt C. Anderson, Sep 29 2014

Keywords

• nonn

Comments

Subset of A134407.

If f(x) = x^2 + 1 and g(c,y) = c*y^2 + y + c then the algebraic substitution of g for x gives a factorization: f(g(c,y)) = (y^2 + 1)*(c^2*y^2 + c^2 + 2*c*y + 1). Since both factors of f(g(c,y)) are integers greater than one, f(g(c,y)) is a composite number.

The numbers are necessarily even terms from A134407 since for odd n = 2c + 1 one has the "forbidden" decomposition with z = 1. - M. F. Hasler, Oct 04 2014

Links

• Eric Weisstein's World of Mathematics, Landau's Problems

Crossrefs

Cf. A134407.

Programs

Maple

maxn:=200:
mb:=proc(n::integer)::integer;
  if isprime(n^2+1)=false then return n else return -1 fi;
end proc:
A134407 := Vector(maxn):
for a from 1 to maxn do A134407[a]:= mb(a): end do:
A134407s:=convert(A134407,'set') minus {-1}:
A134407l:=convert(A134407s,'list'):
for c from 1 to 200 do
  for z from 1 to 20 do
    A134407s := A134407s minus {c*z^2 + z + c}:
  end do:
end do:
A134407s;

PARI

is(n)={!bittest(n,0)&&!isprime(n^2+1)&&!for(z=2,sqrtint(n),(n-z)%(z^2+1)||return)} \\ M. F. Hasler, Oct 04 2014