A373299 Numbers prime(k) such that prime(k) - prime(k-1) = prime(k+2) - prime(k+1).

7, 11, 13, 17, 29, 41, 59, 79, 101, 103, 107, 113, 139, 163, 181, 193, 227, 257, 269, 311, 359, 379, 397, 419, 421, 439, 461, 487, 491, 547, 569, 577, 599, 691, 701, 709, 761, 811, 823, 857, 863, 881, 887, 919, 983, 1021, 1049, 1051, 1091, 1109, 1163

Offset: 1

Author

Alexandre Herrera, May 31 2024

Keywords

• nonn

Examples

			7 is in the list because the prime previous to 7 is 5 and the next primes after 7 are 11 and 13, so we have 7 - 5 = 13 - 11 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001223, A022885, A151800, A263674.

Programs

Maple

P:= select(isprime,[seq(i,i=3..10^4,2)]):
G:= P[2..-1]-P[1..-2]: nG:= nops(G):
J:= select(t -> G[t-1]=G[t+1],[$2..nG-1]):
P[J]; # Robert Israel, May 31 2024

Mathematica

Select[Partition[Prime[Range[200]], 4, 1], #[[2]] - #[[1]] == #[[4]] - #[[3]] &][[;; , 2]] (* Amiram Eldar, May 31 2024 *)

Python

from sympy import prime
def ok(k):
    return prime(k)-prime(k-1) == prime(k+2)-prime(k+1)
print([prime(k) for k in range(2,200) if ok(k)])

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 q-p == s-r: yield q
        p, q, r, s = q, r, s, nextprime(s)
print(list(islice(agen(), 60))) # Michael S. Branicky, May 31 2024

Formula

a(n) = A151800(A022885(n)).