A084165 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.

5, 13, 17, 37, 89, 97, 181, 2689, 2969, 4621, 7457, 8081, 8161, 36709, 62701, 169489, 169709, 169753, 282809, 770101, 5763577, 9491101, 9491281, 9495121, 42544261, 115195501, 189689041, 189689653, 312315373, 312316409, 2294883817

Offset

1,1

Comments

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.

References

Handbook of First Complex Prime Numbers, Part1 + 2 Ervand Kogbetliantz and Alice Krikorian, Gordon and Breach, 1971

Links

Table of n, a(n) for n=1..31.

Example

89 is the 11th prime in sequence A002313, 11*8 = 88, so 89 = 1 mod 11

Mathematica

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 *)

Crossrefs

Cf. A002313, A084163, A084166.

Sequence in context: A248980 A188131 A172459 * A120130 A087484 A019382

Adjacent sequences: A084162 A084163 A084164 * A084166 A084167 A084168

Keyword

nonn

Author

Sven Simon, May 17 2003

Status

approved