A326716 3-term arithmetic progressions of primes whose indices are also primes in arithmetic progression.

5, 11, 17, 461, 617, 773, 401, 599, 797, 877, 1087, 1297, 1471, 1597, 1723, 1217, 1847, 2477, 3001, 3259, 3517, 3001, 3637, 4273, 2417, 3407, 4397, 2081, 3299, 4517, 4339, 4549, 4759, 3733, 4801, 5869, 7193, 8117, 9041, 11927, 12203, 12479, 13103, 13217, 13331

Offset

1,1

Comments

3-term arithmetic progressions are ordered first by highest term, then by middle term, and last by lowest term.

Is there a proof that the sequence is infinite?

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10293

Index entries for sequences related to primes in arithmetic progressions

Formula

a(3*k+2) - a(3*k+1) = a(3*k+3) - a(3*k+2) for k >= 0.

pi(a(3*k+2)) - pi(a(3*k+1)) = pi(a(3*k+3)) - pi(a(3*k+2)) for k >= 0.

a(n) = prime(pi(a(n))) = A000040(A000720(a(n))).

pi(a(n)) = prime(pi(pi(a(n)))).

Example

The indices of 5,11,17 form the arithmetic progression of primes 3,5,7.
The indices of 461,617,773 form the arithmetic progression of primes 89,113,137.

Maple

l:= NULL: nn:= 2000:  # nn = upper limit for index of largest prime found
for n from 3 to nn do
  if isprime(n) then
    for i from iquo(n-1, 2) to 1 by -1 do
      if isprime(n-i) and isprime(n-2*i) then
        p, q, r:= map(ithprime, [seq(n-i*j, j=0..2)])[];
        if p-q = q-r then l:= l, r, q, p
fi fi od fi od: l;  # Alois P. Heinz, Aug 12 2019

Crossrefs

Cf. A000040, A000720, A006450, A231406.

Sequence in context: A271655 A294135 A265851 * A118122 A314256 A314257

Adjacent sequences: A326713 A326714 A326715 * A326717 A326718 A326719

Keyword

nonn,look

Author

Jonathan Sondow, Aug 11 2019

Extensions

More terms from Alois P. Heinz, Aug 12 2019

Status

approved