OFFSET
1,1
COMMENTS
Restricting c and d so that c is congruent to 1 modulo d, we have that the composition of functions k(x) factors. k(x) = (1/d^2)*((1 + x*d^2 + x^2*d^2 - d - 2*x*d + 41*d^2)*(c^2*d^2*x^2 + x*d^2*c^2 + 41*c^2*d^2 + 2*x*d*c^2 - 2*x*d*c^2 + c*d - c^2*d + 1). So k(x) is the product of two integers greater than one and is thus composite.
REFERENCES
John Stillwell, Elements of Number Theory, Springer 2003, page 3.
LINKS
Matt C. Anderson, Table of n, a(n) for n = 1..75
EXAMPLE
If d = 1 then n = c*n^2 + (1 - c)*x + 41*c - 1. This is, up to a change of variables, equivalent to A201998.
MAPLE
maxn := 1000;
A := {};
for n to maxn do
g := n^2+n+41;
if isprime(g) = false then
A := `union`(A, {n}) :
end if :
end do :
A:
# the A list now contains Positive numbers n such that
# n^2 + n + 41 is composite.
# an upper limit for the number of iterations in the
# triple nested while loops is 1000^3 or a billion.
c:=1:
d:=1:
x:=-1:
p:=41:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d;
A2:=A:
while q < maxn do
while q < maxn do
while q < maxn do
A2:=A2 minus {q}:
A2:=A2 minus {c*x^(2)+(c+1)*x+c*p}:
A2:=A2 minus {c*d*x^2-((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d}:
x:=x+1:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
c:=c+1:
x:=-1:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
d:=d+1:
c:=1:
x:=-1:
q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
end do:
A2
CROSSREFS
KEYWORD
nonn
AUTHOR
Matt C. Anderson, Jan 08 2014
EXTENSIONS
Corrected and edited by Matt C. Anderson, Jan 23 2014
STATUS
approved