A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = c*d*x^2 + ((d-2)*c + 1)*x + ((41*d^2 - d + 1)*c -1)/d for an integer x.

611, 622, 630, 663, 679, 734, 758, 835, 867, 966, 978, 995, 1006, 1009, 1060, 1088, 1127, 1142, 1157, 1173, 1175, 1183, 1228, 1280, 1345, 1355, 1368, 1388, 1390, 1426, 1433, 1455, 1457, 1467, 1497, 1538, 1539, 1543, 1554, 1578, 1603, 1612, 1613, 1630, 1661

Offset: 1

Author

Matt C. Anderson, Jan 08 2014

Keywords

• nonn

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.

Examples

			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.
		

References

• John Stillwell, Elements of Number Theory, Springer 2003, page 3.

Links

• Matt C. Anderson, Table of n, a(n) for n = 1..75

Crossrefs

Cf. A007634 (numbers n such that n^2 + n + 41 is composite).

Cf. A201998 and A241529 (similar subsequences of A007634).

Programs

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

Extensions

Corrected and edited by Matt C. Anderson, Jan 23 2014