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%I
%S 0,1,3,13,123,2903,3373867,895293820337
%N a(n) is the smallest number such that a(n)^2 +1 = (a(n-1)^2 + 1)*q(n) where a(0)=0 and q(n) is a sequence of increasing prime numbers.
%e a(0) = 0;
%e a(1) = 1 because a(1)^2+1 = (0^2 + 1)*2 = 2;
%e a(2) = 3 because a(2)^2 + 1 = (1^2 + 1)*5 = 2*5;
%e a(3) = 13 because a(3)^2 + 1 = (3^2+1)*17 =2*5*17;
%e a(4) = 123 because 123^2 + 1 = (13^2 + 1)*89 = 2*5*17*89;
%e a(5) = 2903 because 2903^2 + 1 = (123^2+1)*557 = 2*5*17*89*557;
%e a(6) = 3373867 because 3373867^2+1 = 2 * 5 * 17 * 89 * 557 * 1350709.
%p with(numtheory):nn:=100000:T:=array(1..nn):k:=1: for x from 1 to nn do: p:=4*x+1:if
%p type(p, prime)=true then T[k]:=p:k:=k+1:else fi:od:pp:=2:for u from 1 to k do:id:=0:for
%p x from 1 to nn while(id=0) do:y:=x^2+1:z:=pp*T[u]:if y=z then id:=1:print(x):else
%p fi:od:if id=1 then pp:=y:else fi:od:
%t a[0] = 0; a[1] = 1; lst = {2}; a[n_] := a[n] = (s = 1 + a[n - 1]^2; p = NextPrime[lst[[-1]]]; While[q = Sqrt[p*s - 1]; ! IntegerQ@ q, p = NextPrime@ p]; AppendTo[lst, p]; q) (* _Robert G. Wilson v_, Feb 06 2011 *)
%K nonn,hard
%O 0,3
%A _Michel Lagneau_, Feb 01 2011
%E a(7) from _Donovan Johnson_, Feb 11 2011
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