OFFSET
1,1
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.
FORMULA
a(n) ~ n. - Charles R Greathouse IV, Apr 25 2014
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):
pv2:=Vector(1000, j->((j-1)^2+(j-1))/2+81):
p2:=convert(pv2, set):
ThisSet:=A055390 minus p2 minus {0};
PROG
(PARI) is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) \\ Charles R Greathouse IV, Apr 25 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Matt C. Anderson, Aug 28 2011
EXTENSIONS
Fixed subscript in first comment by Matt C. Anderson
STATUS
approved