cp's OEIS Frontend

A248015 Positive numbers n such that n^2 + 1 is composite and there are no positive integers c and z such that n = c*z^2 + z + c.

%I A248015 #16 Feb 16 2025 08:33:23
%S A248015 8,18,28,30,34,44,46,48,50,58,60,64,68,70,76,78,86,88,96,98,100,104,
%T A248015 108,114,118,128,136,144,148,158,164,166,168,178,186,188,190,194,196,
%U A248015 198,200
%N A248015 Positive numbers n such that n^2 + 1 is composite and there are no positive integers c and z such that n = c*z^2 + z + c.
%C A248015 Subset of A134407.
%C A248015 If f(x) = x^2 + 1 and g(c,y) = c*y^2 + y + c then the algebraic substitution of g for x gives a factorization: f(g(c,y)) = (y^2 + 1)*(c^2*y^2 + c^2 + 2*c*y + 1).  Since both factors of f(g(c,y)) are integers greater than one, f(g(c,y)) is a composite number.
%C A248015 The numbers are necessarily even terms from A134407 since for odd n = 2c + 1 one has the "forbidden" decomposition with z = 1. - _M. F. Hasler_, Oct 04 2014
%H A248015 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LandausProblems.html">Landau's Problems</a>
%p A248015 maxn:=200:
%p A248015 mb:=proc(n::integer)::integer;
%p A248015   if isprime(n^2+1)=false then return n else return -1 fi;
%p A248015 end proc:
%p A248015 A134407 := Vector(maxn):
%p A248015 for a from 1 to maxn do A134407[a]:= mb(a): end do:
%p A248015 A134407s:=convert(A134407,'set') minus {-1}:
%p A248015 A134407l:=convert(A134407s,'list'):
%p A248015 for c from 1 to 200 do
%p A248015   for z from 1 to 20 do
%p A248015     A134407s := A134407s minus {c*z^2 + z + c}:
%p A248015   end do:
%p A248015 end do:
%p A248015 A134407s;
%o A248015 (PARI) is(n)={!bittest(n,0)&&!isprime(n^2+1)&&!for(z=2,sqrtint(n),(n-z)%(z^2+1)||return)} \\ _M. F. Hasler_, Oct 04 2014
%Y A248015 Cf. A134407.
%K A248015 nonn
%O A248015 1,1
%A A248015 _Matt C. Anderson_, Sep 29 2014