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%I #8 Nov 20 2019 13:43:35
%S 1,3,7,13,21,31,43,91,111,183,211,241,273,381,421,553,601,651,703,
%T 1261,1333,1561,1641,2863,2971,3081,3193,4291,4423,5403,5551,6973,
%U 7141,8011,8191,8743,8931,11991,12211,13341,13573,14281,14521,15253,15501,15751,16003
%N Numbers n such that the greatest prime divisor p of n^2+1 has the property that (p - n)^2 + 1 = p.
%C For the n > 1 in this sequence, n^2+1 is composite. The corresponding primes p are A002496(n) repeated two times for n > 1 : {2, 5, 5, 17, 17, 37, 37, 101, 101, 197,...}.
%C Because this sequence is connected with A002496, it is conjectured that the set of this numbers is infinite.
%e 31 is in the sequence because 31^2 + 1 = 2*13*37 and (37 - 31)^2 + 1 = 37.
%e 43 is in the sequence because 43^2 + 1 = 2*5*5*37 and (37 - 43)^2 + 1 = 37.
%p with(numtheory):for n from 1 to 20000 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):p:=y[n1]:q:=(p-n)^2+1:if q=p then printf(`%d, `,n): else fi:od:
%t pn2pQ[n_]:=Module[{p=FactorInteger[n^2+1][[-1,1]]},(p-n)^2+1==p]; Select[ Range[20000],pn2pQ] (* _Harvey P. Dale_, Nov 20 2019 *)
%Y Cf. A002496, A134406.
%K nonn
%O 1,2
%A _Michel Lagneau_, Feb 05 2012
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