A194565 - OEIS

%I #20 Apr 25 2014 10:28:54

%S 122,123,127,130,138,143,155,162,163,164,170,173,178,185,187,190,204,

%T 205,207,208,213,215,216,218,232,237,239,242,244,245,246,248,249,251,

%U 255,256,259,261,266,268,270,278,279,283,284,286,287,289,295,298,299,300

%N 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.

%C 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.

%C 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.

%C 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.

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

%H Charles R Greathouse IV, <a href="/A194565/b194565.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">MathWorld: Prime-Generating Polynomial</a>.

%F a(n) ~ n. - _Charles R Greathouse IV_, Apr 25 2014

%p AV:=Vector(1000,0):

%p counter:=1:

%p for n from 1 to 1000 do

%p g:=n^2+n+41:

%p if isprime(g)=false then

%p   AV[counter]:=n:

%p   counter:=counter+1:

%p end if

%p end do:

%p A007634:=convert(AV,set):

%p pv1:=Vector(1000,j->(j-1)^2+40):

%p p1:=convert(pv1,set):

%p A055390:=A007634 minus p1:

%p pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):

%p p2:=convert(pv2,set):

%p ThisSet:=A055390 minus p2 minus {0};

%o (PARI) is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) \\ _Charles R Greathouse IV_, Apr 25 2014

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

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

%K nonn,easy

%O 1,1

%A _Matt C. Anderson_, Aug 28 2011

%E Fixed subscript in first comment by Matt C. Anderson