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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.

%I A194634 #24 Feb 16 2025 08:33:15
%S A194634 127,138,155,163,164,170,173,178,185,190,204,205,207,208,213,215,216,
%T A194634 232,237,239,242,244,245,246,248,249,251,256,259,261,266,268,270,278,
%U A194634 279,283,284,286,287,289,295,299,300,301,302,309,314,321,325,326,327,328
%N 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.
%C A194634 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.
%C A194634 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.
%C A194634 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.
%C A194634 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.
%C A194634 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.
%D A194634 John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
%D A194634 R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective 2nd ed., Springer, 2005, page 21.
%H A194634 Charles R Greathouse IV, <a href="/A194634/b194634.txt">Table of n, a(n) for n = 1..10000</a>
%H A194634 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%F A194634 a(n) ~ n. - _Charles R Greathouse IV_, Apr 25 2014
%p A194634 A007634:={}:
%p A194634 for n from 1 to 1000 do
%p A194634 k:=n^2+n+41:
%p A194634 if isprime(k)=false then
%p A194634 A007634:=A007634 union {n}:
%p A194634 end if:
%p A194634 end do:
%p A194634 pv1:=Vector(1000,j->(j-1)^2+40):
%p A194634 p1:=convert(pv1,set):
%p A194634 A055390:=A007634 minus p1 minus {0}:
%p A194634 pv2:=Vector(1000,j->((j-1)^2+(j-1))/2+81):
%p A194634 p2:=convert(pv2,set):
%p A194634 A194565:=A055390 minus p2:
%p A194634 pv3:=Vector(1000,j->(3*(j-1)^2-2*(j-1)+122)):
%p A194634 p3:=convert(pv3,set):
%p A194634 p3set:=A194565 minus p3;
%o A194634 (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
%Y A194634 Cf. A007634 (n such that n^2+n+41 is composite).
%Y A194634 Cf. A055390 (members of A007634 that are not lattice points of x^2+40).
%Y A194634 Cf. A194565 (members of A055390 that are not lattice points of (x^2+x)/2 + 81).
%K A194634 nonn,easy
%O A194634 1,1
%A A194634 _Matt C. Anderson_, Aug 30 2011
%E A194634 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.