A201998 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = c*x^2 + (c + 1)*x + c*41 for an integer x.

244, 249, 251, 266, 270, 295, 301, 336, 344, 389, 399, 407, 416, 418, 445, 449, 454, 466, 489, 494, 496, 500, 506, 527, 531, 545, 547, 563, 570, 571, 582, 583, 585, 611, 612, 620, 622, 624, 628, 630, 636, 652, 661, 662, 663, 679, 693, 699

Offset: 1

Author

Matt C. Anderson, Dec 07 2011

Keywords

• nonn

Comments

The composition of functions k(x) factors. k(x) = (x^2 + x + 41)*(c^2*x^2 + (c^2 + 2*c)*x + c^2*41 + c + 1). So k(x) is the product of two integers greater than one and thus composite.

References

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

Links

• Matt C. Anderson A prime producing polynomial writeup

Crossrefs

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

Cf. A235381 (similar to this sequence).

Programs

Maple

maxn:=1000:
A:={}:
for n from 1 to maxn do
g:=n^2+n+41:
if isprime(g)=false then
A:=A union {n}:
end if:
end do:
# The set A contains values n such that n^2+n+41 is composite and n < maxn.
c:=1:
x:=-1:
p:=41:
q:=c*x^2-(c+1)*x+c*p:
A2:=A:
while q < maxn do
while q < maxn do
A2:=A2 minus {q}:
A2:=A2 minus {c*x^2+(c+1)*x+c*p}:
x:=x+1:
q:=c*x^2-(c+1)*x+c*p:
end do:
c:=c+1:
x:=-1:
q:=c*x^2-(c+1)*x+c*p:
end do:
A2;

Mathematica

Reap[For[n=1, n<700, n++, If[!PrimeQ[n^2+n+41], If[Reduce[c>0 && n == c*x^2+(c+1)*x+41*c , {c, x}, Integers] === False, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Apr 30 2014 *)