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 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. 0
 8, 18, 28, 30, 34, 44, 46, 48, 50, 58, 60, 64, 68, 70, 76, 78, 86, 88, 96, 98, 100, 104, 108, 114, 118, 128, 136, 144, 148, 158, 164, 166, 168, 178, 186, 188, 190, 194, 196, 198, 200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subset of A134407. 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. 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 LINKS Table of n, a(n) for n=1..41. Eric Weisstein's World of Mathematics, Landau's Problems MAPLE maxn:=200: mb:=proc(n::integer)::integer; if isprime(n^2+1)=false then return n else return -1 fi; end proc: A134407 := Vector(maxn): for a from 1 to maxn do A134407[a]:= mb(a): end do: A134407s:=convert(A134407, 'set') minus {-1}: A134407l:=convert(A134407s, 'list'): for c from 1 to 200 do for z from 1 to 20 do A134407s := A134407s minus {c*z^2 + z + c}: end do: end do: A134407s; PROG (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 CROSSREFS Cf. A134407. Sequence in context: A290614 A347640 A347995 * A011538 A283611 A043521 Adjacent sequences: A248012 A248013 A248014 * A248016 A248017 A248018 KEYWORD nonn AUTHOR Matt C. Anderson, Sep 29 2014 STATUS approved

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Last modified October 3 09:32 EDT 2023. Contains 365854 sequences. (Running on oeis4.)