login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A084161 Primes that are the sum of two squares and which set a record for the gap to the next prime of that form. 7
2, 5, 17, 73, 113, 197, 461, 1493, 1801, 9533, 15661, 16741, 33181, 39581, 50593, 180797, 183089, 1561829, 1637813, 2243909, 4468889, 4874717, 7856441, 10087201, 12021029, 12213913, 18226661, 148363637, 292182097, 320262253, 468213937 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Real primes 2, 5, 13, 17, 29, 37, ... (A002313) have a unique representation as sum of two squares. Values larger than 2 are the primes p with p = 1 mod 4. If p = x^2 + y^2, the corresponding complex prime is x + y * i, where i is the imaginary unit.

The length of the gap can be found in A084162.

REFERENCES

Ervand Kogbetliantz and Alice Krikorian, Handbook of First Complex Prime Numbers, Parts 1 and 2, Gordon and Breach, 1971.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..42

Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

EXAMPLE

a(3) = 73: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the length is 16. Note that 73 = (8 - 3i)(8 + 3i) and 89 = (8 - 5i)(8 + 5i). The primes 79 and 83 are inert in Z[i].

MATHEMATICA

Reap[Print[2]; Sow[2]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[p] Sow[p]]; p = q]][[2, 1]] (* Jean-Fran├žois Alcover, Feb 20 2019, from PARI *)

PROG

(PARI) print1(2); r=0; p=5; forprime(q=7, 1e7, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "p)); p=q) \\ Charles R Greathouse IV, Apr 29 2014

CROSSREFS

Cf. A002313, A084160, A084162 (gap sizes), A268963 (end-of-gap primes).

Sequence in context: A104859 A108289 A007779 * A325294 A230960 A102038

Adjacent sequences:  A084158 A084159 A084160 * A084162 A084163 A084164

KEYWORD

nonn

AUTHOR

Sven Simon, May 17 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 28 13:24 EDT 2020. Contains 337393 sequences. (Running on oeis4.)