A391116 Prime-indexed primes that are the average of the preceding and following prime-indexed primes.

11, 431, 2081, 2683, 3169, 4663, 25841, 26591, 76031, 92569, 127033, 158227, 158293, 158449, 186481, 204019, 213067, 218513, 221539, 291437, 294313, 397013, 419351, 426841, 510481, 551443, 591937, 594203, 609313, 671303, 677227, 756227, 782137, 822667, 854353

Offset

1,1

Comments

Middle terms of three consecutive prime-indexed primes that form an arithmetic progression.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

Terms q(k) such that 2*q(k) = q(k-1) + q(k+1) where q(n) = prime(prime(n)) = A006450(n).

Example

11 is a term because A006450 contains 5, 11, 17 and 11 = (5 + 17) / 2.
431 is a term because A006450 contains 419, 431, 443 and 431 = (419 + 443) / 2.

Mathematica

With[{pip = Prime[Prime[Range[10000]]]}, pip[[1 + Position[Differences[pip, 2], 0] // Flatten]]] (* Amiram Eldar, Dec 01 2025 *)

Prog

(Python)
from sympy import prime
def pip(n):
    return prime(prime(n))
def equidistant_pips(max_k=10000):
    terms = []
    indices = []
    q_prev = pip(1)
    q_curr = pip(2)
    for k in range(2, max_k):
        q_next = pip(k + 1)
        if q_curr - q_prev == q_next - q_curr:
            terms.append(q_curr)
            indices.append(k)
        q_prev, q_curr = q_curr, q_next
    return terms, indices
(PARI) lista(n)={my(L=List(), k=0, q=0, r=0); forprime(p=2, oo, k++; if(isprime(k), if(p-q==q-r, listput(L, q); if(#L==n, break)); r=q; q=p)); Vec(L)} \\ Andrew Howroyd, Nov 29 2025

Crossrefs

Subsequence of A006450, similar to A006562.

Sequence in context: A356210 A140840 A175158 * A360066 A354439 A180087

Adjacent sequences: A391113 A391114 A391115 * A391117 A391118 A391119

Keyword

nonn

Author

Bruce Nye, Nov 29 2025

Status

approved