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%I #8 Aug 13 2020 12:52:01
%S 5,13,17,37,89,97,181,2689,2969,4621,7457,8081,8161,36709,62701,
%T 169489,169709,169753,282809,770101,5763577,9491101,9491281,9495121,
%U 42544261,115195501,189689041,189689653,312315373,312316409,2294883817
%N Primes which are 1 mod m, where m is the index of the prime in sequence A002313 (Real primes with corresponding complex primes). The index m can be found in A084166 Primes which are -1 mod m can be found in sequence A084163.
%C Real primes 2,5,13,17,29,37,... have a unique representation as sum of two squares. Values larger 2 are the primes p with p = 1 mod 4. This is sequence A002313. If p = x^2 + y^2, the corresponding complex prime is x+y*i First complex prime is 1+i with 2 as corresponding real prime, according to reference, page 1-2.
%D Handbook of First Complex Prime Numbers, Part1 + 2 Ervand Kogbetliantz and Alice Krikorian, Gordon and Breach, 1971
%e 89 is the 11th prime in sequence A002313, 11*8 = 88, so 89 = 1 mod 11
%t Module[{nn=112*10^6,pr,len},pr=Select[Prime[Range[nn]],MemberQ[ {1,2},Mod[ #,4]]&];len=Length[pr];Select[Thread[{pr,Range[len]}],Mod[ #[[1]],#[[2]]] == 1&]][[All,1]] (* _Harvey P. Dale_, Aug 13 2020 *)
%Y Cf. A002313, A084163, A084166.
%K nonn
%O 1,1
%A _Sven Simon_, May 17 2003
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