A194634 Numbers n such that k= n^2 + n + 41 is composite and there is no integer x such that n= x^2 + 40; n= (x^2+x)/2 + 81; or n= 3*x^2 - 2x + 122.

127, 138, 155, 163, 164, 170, 173, 178, 185, 190, 204, 205, 207, 208, 213, 215, 216, 232, 237, 239, 242, 244, 245, 246, 248, 249, 251, 256, 259, 261, 266, 268, 270, 278, 279, 283, 284, 286, 287, 289, 295, 299, 300, 301, 302, 309, 314, 321, 325, 326, 327, 328

Offset: 1

Author

Matt C. Anderson, Aug 30 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. A third fit: p3(0)=122; p3(1)=123; p3(2)=130 yields p3(x)=3x^2-2*x+122.

Substituting n=x^2 into k=n^2+n+41 is factorable as: k1=(x^2+x+41)*(x^2-x+41). This shows that all lattice points on p1 produce a composite k.

Similarly, substituting n=(x^2-x)/2 + 81 into k factors as k2=(x^2+163)*(x^2+2*x+164)/4. So all lattice points on p2 produce a composite k.

Similarly, substituting n=3*x^2-2*x+122 into k factors as k3=(x^2-x+41)*(9*x^2-3*x+367). So all lattice points on p3 produce a composite k.

This procedure can be continued with p4(x)=3*x^2+8*x+127, p5(x)=4*x^2-3*x+163, p6(x)=4*x^2+11*x+170, p7(x)=5*x^2-4*x+204, p8(x)=5*x^2+14*x+213, p9(x)=(3*x^2-x)/2+244, p10(x)=(3*x^2+7*x)/2+246, and so on.

References

• John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
• R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective 2nd ed., Springer, 2005, page 21.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, 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).

Cf. A194565 (members of A055390 that are not lattice points of (x^2+x)/2 + 81).

Programs

Maple

A007634:={}:
for n from 1 to 1000 do
k:=n^2+n+41:
if isprime(k)=false then
A007634:=A007634 union {n}:
end if:
end do:
pv1:=Vector(1000,j->(j-1)^2+40):
p1:=convert(pv1,set):
A055390:=A007634 minus p1 minus {0}:
pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
p2:=convert(pv2,set):
A194565:=A055390 minus p2:
pv3:=Vector(1000,j->(3*(j-1)^2-2*(j-1)+122)):
p3:=convert(pv3,set):
p3set:=A194565 minus p3;

PARI

is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) && n > 126 && (x->3*x^2-2*x+122)(round((1+sqrt(3*n-365))/3))!=n \\ Charles R Greathouse IV, Apr 25 2014

Formula

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

Extensions

Fixed subscripts in first comment. Added * in 4th comment. Added 5th comment. Changed g to k for consistancy. Improved Maple code. Added second book reference. Changed name to agree with comment of editor.