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%I #27 May 15 2014 09:57:06
%S 244,249,251,266,270,295,301,336,344,389,399,407,416,418,445,449,454,
%T 466,489,494,496,500,506,527,531,545,547,563,570,571,582,583,585,611,
%U 612,620,622,624,628,630,636,652,661,662,663,679,693,699
%N Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = c*x^2 + (c + 1)*x + c*41 for an integer x.
%C The composition of functions k(x) factors. k(x) = (x^2 + x + 41)*(c^2*x^2 + (c^2 + 2*c)*x + c^2*41 + c + 1). So k(x) is the product of two integers greater than one and thus composite.
%D John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
%H Matt C. Anderson <a href="https://sites.google.com/site/mattc1anderson/home-1">A prime producing polynomial writeup</a>
%p maxn:=1000:
%p A:={}:
%p for n from 1 to maxn do
%p g:=n^2+n+41:
%p if isprime(g)=false then
%p A:=A union {n}:
%p end if:
%p end do:
%p # The set A contains values n such that n^2+n+41 is composite and n < maxn.
%p c:=1:
%p x:=-1:
%p p:=41:
%p q:=c*x^2-(c+1)*x+c*p:
%p A2:=A:
%p while q < maxn do
%p while q < maxn do
%p A2:=A2 minus {q}:
%p A2:=A2 minus {c*x^2+(c+1)*x+c*p}:
%p x:=x+1:
%p q:=c*x^2-(c+1)*x+c*p:
%p end do:
%p c:=c+1:
%p x:=-1:
%p q:=c*x^2-(c+1)*x+c*p:
%p end do:
%p A2;
%t Reap[For[n=1, n<700, n++, If[!PrimeQ[n^2+n+41], If[Reduce[c>0 && n == c*x^2+(c+1)*x+41*c , {c, x}, Integers] === False, Sow[n]]]]][[2, 1]] (* _Jean-François Alcover_, Apr 30 2014 *)
%Y Cf. A007634 (n^2 + n + 41 is composite).
%Y Cf. A235381 (similar to this sequence).
%K nonn
%O 1,1
%A _Matt C. Anderson_, Dec 07 2011
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