OFFSET
1,1
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
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;
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Matt C. Anderson, Sep 29 2014
STATUS
approved