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A386516 Least k such that k^2+1 contains exactly n distinct prime factors of the form m^2+1 or 0 if no such k exists.

%I A386516 #25 Jul 25 2025 14:23:45
%S A386516 1,3,13,183,2843,41323,57109753,1929510527,760999599793
%N A386516 Least k such that k^2+1 contains exactly n distinct prime factors of the form m^2+1 or 0 if no such k exists.
%e A386516 a(4)=183 because the prime factors of 183^2+1 are {2, 5, 17, 197} are of the form m^2+1 with m = 1, 2, 4 and 14.
%p A386516 with(numtheory):nn:=10^6:
%p A386516 for n from 0 to 6 do :
%p A386516  ii :=0 :
%p A386516  for k from 0 to nn while(ii=0) do :
%p A386516    d:=factorset(k^2+1):n0:=nops(d):it:=0:
%p A386516     for i from 1 to n0 do:
%p A386516       c:=d[i]-1:if sqrt(c) = floor(sqrt(c)) then it:=it+1:else fi:
%p A386516     od:
%p A386516      if it =n then ii :=1 :printf (`%d %d \n`,n,k):
%p A386516       else
%p A386516      fi :
%p A386516  od:
%p A386516 od :
%t A386516 a[n_]:=Module[{k=0},Until[PrimeNu[k^2+1]==n&&AllTrue[Sqrt[First/@FactorInteger[k^2+1]-1],IntegerQ],k++];k];Array[a,6] (* _James C. McMahon_, Jul 25 2025 *)
%Y A386516 Cf. A002522, A001912, A002496, A005574, A008784, A216784.
%K A386516 nonn,more
%O A386516 1,2
%A A386516 _Michel Lagneau_, Jul 24 2025
%E A386516 a(7) from _Giovanni Resta_, Jul 24 2025
%E A386516 a(8)-a(9) from _David A. Corneth_, Jul 24 2025