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

%I A326716 #37 Aug 13 2019 16:34:54
%S A326716 5,11,17,461,617,773,401,599,797,877,1087,1297,1471,1597,1723,1217,
%T A326716 1847,2477,3001,3259,3517,3001,3637,4273,2417,3407,4397,2081,3299,
%U A326716 4517,4339,4549,4759,3733,4801,5869,7193,8117,9041,11927,12203,12479,13103,13217,13331
%N A326716 3-term arithmetic progressions of primes whose indices are also primes in arithmetic progression.
%C A326716 3-term arithmetic progressions are ordered first by highest term, then by middle term, and last by lowest term.
%C A326716 Is there a proof that the sequence is infinite?
%H A326716 Alois P. Heinz, <a href="/A326716/b326716.txt">Table of n, a(n) for n = 1..10293</a>
%H A326716 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F A326716 a(3*k+2) - a(3*k+1) = a(3*k+3) - a(3*k+2) for k >= 0.
%F A326716 pi(a(3*k+2)) - pi(a(3*k+1)) = pi(a(3*k+3)) - pi(a(3*k+2)) for k >= 0.
%F A326716 a(n) = prime(pi(a(n))) = A000040(A000720(a(n))).
%F A326716 pi(a(n)) = prime(pi(pi(a(n)))).
%e A326716 The indices of 5,11,17 form the arithmetic progression of primes 3,5,7.
%e A326716 The indices of 461,617,773 form the arithmetic progression of primes 89,113,137.
%p A326716 l:= NULL: nn:= 2000:  # nn = upper limit for index of largest prime found
%p A326716 for n from 3 to nn do
%p A326716   if isprime(n) then
%p A326716     for i from iquo(n-1, 2) to 1 by -1 do
%p A326716       if isprime(n-i) and isprime(n-2*i) then
%p A326716         p, q, r:= map(ithprime, [seq(n-i*j, j=0..2)])[];
%p A326716         if p-q = q-r then l:= l, r, q, p
%p A326716 fi fi od fi od: l;  # _Alois P. Heinz_, Aug 12 2019
%Y A326716 Cf. A000040, A000720, A006450, A231406.
%K A326716 nonn,look
%O A326716 1,1
%A A326716 _Jonathan Sondow_, Aug 11 2019
%E A326716 More terms from _Alois P. Heinz_, Aug 12 2019