A194565 Numbers n such that n^2 + n + 41 is composite and n is not a lattice point on the parabolas p1 = x^2 + 40 or p2 = (x^2+x)/2 + 81.

122, 123, 127, 130, 138, 143, 155, 162, 163, 164, 170, 173, 178, 185, 187, 190, 204, 205, 207, 208, 213, 215, 216, 218, 232, 237, 239, 242, 244, 245, 246, 248, 249, 251, 255, 256, 259, 261, 266, 268, 270, 278, 279, 283, 284, 286, 287, 289, 295, 298, 299, 300

Offset: 1

Author

Matt C. Anderson, Aug 28 2011

Keywords

• nonn
• easy

Comments

The parabola curve fit: p1(0)=40; p1(1)=41; p1(2)=44 yields p1(x)=x^2+40. A second fit: p2(0)=81; p2(1)=82; p2(2)=84 yields p2(x)=(x^2+x)/2 + 81.

Substituting n=x^2+40 into g=n^2+n+41 is factorable as: g1=(x^2+x+41)*(x^2-x+41). This shows that all lattice points on p1 produce a composite g.

Similarly, substituting n=(x^2-x)/2 + 81 into g factors as g2=(x^2+163)*(x^2+2*x+164)/4. So all lattice points on p2 produce a composite g.

References

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric W. Weisstein, MathWorld: Prime-Generating Polynomial.

Crossrefs

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

Cf. A055390 (members of A007634 that are not lattice points of x^2+40).

Programs

Maple

AV:=Vector(1000,0):
counter:=1:
for n from 1 to 1000 do
g:=n^2+n+41:
if isprime(g)=false then
  AV[counter]:=n:
  counter:=counter+1:
end if
end do:
A007634:=convert(AV,set):
pv1:=Vector(1000,j->(j-1)^2+40):
p1:=convert(pv1,set):
A055390:=A007634 minus p1:
pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
p2:=convert(pv2,set):
ThisSet:=A055390 minus p2 minus {0};

PARI

is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) \\ Charles R Greathouse IV, Apr 25 2014

Formula

a(n) ~ n. - Charles R Greathouse IV, Apr 25 2014

Extensions

Fixed subscript in first comment by Matt C. Anderson