A084162 -id:A084162 - cp's OEIS frontend

A268963 Primes 4k+1 at the end of the maximal gaps in A084162.

5, 13, 29, 89, 137, 229, 509, 1549, 1861, 9601, 15733, 16829, 33289, 39709, 50741, 180949, 183289, 1562053, 1638053, 2244157, 4469141, 4874977, 7856713, 10087481, 12021353, 12214273, 18227081, 148364081, 292182557, 320262769, 468214457, 727335397, 869766761

Offset: 0

Alexei Kourbatov, Feb 16 2016

nonn

Subsequence of A002144.

A084161 lists the primes preceding the maximal gaps, and A084162 lists the corresponding gap sizes. See more comments there.

			a(3) = 89: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the gap size is 16.

Alexei Kourbatov, Table of n, a(n) for n = 0..42

Cf. A002144, A002313, A084161, A084162.

Reap[Print[5]; Sow[5]; 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[q] Sow[q]]; p = q]][[2, 1]] (* Jean-François Alcover, Feb 20 2019, from PARI *)

print1(5); r=0; p=5; forprime(q=7, 1e9, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "q)); p=q)

a(n) = A084161(n) + A084162(n)

A084161 Primes that are the sum of two squares and which set a record for the gap to the next prime of that form.

Original entry on oeis.org

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

Offset: 0

Sven Simon, May 17 2003

nonn

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.

			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].

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

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.

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

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-François Alcover, Feb 20 2019, from 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

cp's OEIS Frontend

A268963 Primes 4k+1 at the end of the maximal gaps in A084162.

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