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A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = c*d*x^2 + ((d-2)*c + 1)*x + ((41*d^2 - d + 1)*c -1)/d for an integer x.

%I A235381 #27 Sep 09 2014 13:24:11
%S A235381 611,622,630,663,679,734,758,835,867,966,978,995,1006,1009,1060,1088,
%T A235381 1127,1142,1157,1173,1175,1183,1228,1280,1345,1355,1368,1388,1390,
%U A235381 1426,1433,1455,1457,1467,1497,1538,1539,1543,1554,1578,1603,1612,1613,1630,1661
%N A235381 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c or d such that n = c*d*x^2 + ((d-2)*c + 1)*x + ((41*d^2 - d + 1)*c -1)/d for an integer x.
%C A235381 Restricting c and d so that c is congruent to 1 modulo d, we have that the composition of functions k(x) factors. k(x) = (1/d^2)*((1 + x*d^2 + x^2*d^2 - d - 2*x*d + 41*d^2)*(c^2*d^2*x^2 + x*d^2*c^2 + 41*c^2*d^2 + 2*x*d*c^2 - 2*x*d*c^2 + c*d - c^2*d + 1). So k(x) is the product of two integers greater than one and is thus composite.
%D A235381 John Stillwell, Elements of Number Theory, Springer 2003, page 3.
%H A235381 Matt C. Anderson, <a href="/A235381/b235381.txt">Table of n, a(n) for n = 1..75</a>
%e A235381 If d = 1 then n = c*n^2 + (1 - c)*x + 41*c  - 1. This is, up to a change of variables, equivalent to A201998.
%p A235381 maxn := 1000;
%p A235381 A := {};
%p A235381 for n to maxn do
%p A235381 g := n^2+n+41;
%p A235381 if isprime(g) = false then
%p A235381 A := `union`(A, {n}) :
%p A235381 end if :
%p A235381 end do :
%p A235381 A:
%p A235381 # the A list now contains Positive numbers n such that
%p A235381 # n^2 + n + 41 is composite.
%p A235381 # an upper limit for the number of iterations in the
%p A235381 # triple nested while loops is 1000^3 or a billion.
%p A235381 c:=1:
%p A235381 d:=1:
%p A235381 x:=-1:
%p A235381 p:=41:
%p A235381 q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d;
%p A235381 A2:=A:
%p A235381 while q < maxn do
%p A235381 while q < maxn do
%p A235381 while q < maxn do
%p A235381   A2:=A2 minus {q}:
%p A235381   A2:=A2 minus {c*x^(2)+(c+1)*x+c*p}:
%p A235381   A2:=A2 minus {c*d*x^2-((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d}:
%p A235381   x:=x+1:
%p A235381   q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
%p A235381 end do:
%p A235381 c:=c+1:
%p A235381 x:=-1:
%p A235381 q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
%p A235381 end do:
%p A235381 d:=d+1:
%p A235381 c:=1:
%p A235381 x:=-1:
%p A235381 q:=c*d*x^2+((d-2)*c+1)*x+((p*d^2-d+1)*c-1)/d:
%p A235381 end do:
%p A235381 A2
%Y A235381 Cf. A007634 (numbers n such that n^2 + n + 41 is composite).
%Y A235381 Cf. A201998 and A241529 (similar subsequences of A007634).
%K A235381 nonn
%O A235381 1,1
%A A235381 _Matt C. Anderson_, Jan 08 2014
%E A235381 Corrected and edited by _Matt C. Anderson_, Jan 23 2014