A022885 Primes p=prime(k) such that prime(k) + prime(k+3) = prime(k+1) + prime(k+2).

5, 7, 11, 13, 23, 37, 53, 73, 97, 101, 103, 109, 137, 157, 179, 191, 223, 251, 263, 307, 353, 373, 389, 409, 419, 433, 457, 479, 487, 541, 563, 571, 593, 683, 691, 701, 757, 809, 821, 853, 859, 877, 883, 911, 977, 1019, 1039, 1049, 1087, 1103

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

These are primes p for which the subsequent alternate prime gaps are equal, so (p(k+3)-p(k+2))/(p(k+1)-p(k)) = 1. It is conjectured that the most frequent alternate prime gaps ratio is one. - Andres Cicuttin, Nov 07 2016

Examples

			Starting from 5, the four consecutive primes are 5, 7, 11, 13; and they satisfy 5 + 13 = 7 + 11. So 5 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 Vincenzo Librandi)

Crossrefs

Cf. A022884, A260179, A261470.

Programs

Magma

[NthPrime(n): n in [1..200] | (NthPrime(n)+NthPrime(n+3)) eq (NthPrime(n+1)+NthPrime(n+2))]; // Vincenzo Librandi, Nov 08 2016

Mathematica

Transpose[Select[Partition[Prime[Range[500]],4,1],First[#]+Last[#] == #[[2]]+#[[3]]&]][[1]] (* Harvey P. Dale, May 23 2011 *)

PARI

isok(p) = {my(k = primepi(p)); (p == prime(k)) && ((prime(k) + prime(k+3)) == (prime(k+1) + prime(k+2)));} \\ Michel Marcus, Jan 15 2014

Python

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    p, q, r, s = [2, 3, 5, 7]
    while True:
        if p + s == q + r: yield p
        p, q, r, s = q, r, s, nextprime(s)
print(list(islice(agen(), 50))) # Michael S. Branicky, May 31 2024

Formula

a(n) = A000040(A022884(n)). - Amiram Eldar, May 06 2020

Extensions

Name edited by Michel Marcus, Jan 15 2014