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

Author

Jonathan Sondow, Aug 11 2019

Keywords

• nonn
• look

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?

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10293
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A000040, A000720, A006450, A231406.

Programs

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

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)))).

Extensions

More terms from Alois P. Heinz, Aug 12 2019