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A326716
3-term arithmetic progressions of primes whose indices are also primes in arithmetic progression.
1
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?
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
KEYWORD
nonn,look
AUTHOR
Jonathan Sondow, Aug 11 2019
EXTENSIONS
More terms from Alois P. Heinz, Aug 12 2019
STATUS
approved