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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.

%I A194565 #21 Feb 16 2025 08:33:15
%S A194565 122,123,127,130,138,143,155,162,163,164,170,173,178,185,187,190,204,
%T A194565 205,207,208,213,215,216,218,232,237,239,242,244,245,246,248,249,251,
%U A194565 255,256,259,261,266,268,270,278,279,283,284,286,287,289,295,298,299,300
%N 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.
%C A194565 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 A194565 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 A194565 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 A194565 John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
%H A194565 Charles R Greathouse IV, <a href="/A194565/b194565.txt">Table of n, a(n) for n = 1..10000</a>
%H A194565 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">MathWorld: Prime-Generating Polynomial</a>.
%F A194565 a(n) ~ n. - _Charles R Greathouse IV_, Apr 25 2014
%p A194565 AV:=Vector(1000,0):
%p A194565 counter:=1:
%p A194565 for n from 1 to 1000 do
%p A194565 g:=n^2+n+41:
%p A194565 if isprime(g)=false then
%p A194565   AV[counter]:=n:
%p A194565   counter:=counter+1:
%p A194565 end if
%p A194565 end do:
%p A194565 A007634:=convert(AV,set):
%p A194565 pv1:=Vector(1000,j->(j-1)^2+40):
%p A194565 p1:=convert(pv1,set):
%p A194565 A055390:=A007634 minus p1:
%p A194565 pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
%p A194565 p2:=convert(pv2,set):
%p A194565 ThisSet:=A055390 minus p2 minus {0};
%o A194565 (PARI) is(n)=!isprime(n^2+n+41) && !issquare(n-40) && !issquare(8*n-647) \\ _Charles R Greathouse IV_, Apr 25 2014
%Y A194565 Cf. A007634 (n such that n^2+n+41 is composite).
%Y A194565 Cf. A055390 (members of A007634 that are not lattice points of x^2+40).
%K A194565 nonn,easy
%O A194565 1,1
%A A194565 _Matt C. Anderson_, Aug 28 2011
%E A194565 Fixed subscript in first comment by Matt C. Anderson