A231406 -id:A231406 - cp's OEIS frontend

A231017 Least prime q > p = prime(n) such that if d = q-p, then p, p+d, p+2d, ..., p+(p-1)d are all primes.

3, 5, 11, 157, 1536160091, 9918821194603, 341976204789992332577, 2166703103992332274919569

Offset: 1

Jonathan Sondow, Nov 08 2013

Second term in the smallest non-constant p-term arithmetic progression (AP) of primes beginning with p = prime(n).
For any non-constant AP beginning with a prime p and ending with a prime, the maximum possible length is p, since p+pd is not prime for d > 0. If all the terms are prime, then the common difference d must be a multiple of all primes < p.
Ribenboim says that in 1986 G. Loh [Loeh] discovered a(5) and a(6), and that a(n) should exist for all n, but "in my opinion, this is so difficult that no one will prove [it], and no one will find a counterexample in the near future."
Phil Carmody found a(7) in 2001.
See the crossrefs for more comments, references, and links.

			Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = 11.

P. Ribenboim, My Numbers, My Friends, Springer, 2000; p. 67.
P. Ribenboim, The Book of Prime Number Records, 2nd ed., Springer, 1989; p. 225.

Phil Carmody, a(7), NMBRTHRY November 2001.
Index entries for sequences related to primes in arithmetic progressions

For common differences see A088430, for initial terms see A000040, for last terms see A113834, for the APs see A231406.

For other types of APs of primes see A005115 and its comments.

a(n)=my(p=prime(n),P=prod(i=1,n-1,prime(i)),d); forprime(q=p+1,, d=q-p; if(d%P,next); for(i=2,p-1,if(!isprime(p+i*d), next(2))); return(q)) \\ Charles R Greathouse IV, Nov 08 2013

a(n) = prime(n) + A088430(n) = prime(n) + A002110(n)*A231018(n).

a(8) found by Wojciech Izykowski, from Jens Kruse Andersen, Jun 30 2014

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

Original entry on oeis.org

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

Jonathan Sondow, Aug 11 2019

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?

			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.

Cf. A000040, A000720, A006450, A231406.

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

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)))).

More terms from Alois P. Heinz, Aug 12 2019

A133278 Triangle read by rows, with n-th row the smallest non-constant n-term arithmetic progression of primes beginning with prime(n).

Original entry on oeis.org

2, 3, 5, 5, 11, 17, 7, 19, 31, 43

Offset: 1

N. J. A. Sloane, Oct 17 2007

			Triangle begins:
2
3 5
5 11 17
7 19 31 43

Index entries for sequences related to primes in arithmetic progressions

Cf. A231406.

Sequence corrected by Jonathan Sondow, Nov 08 2013

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A231017 Least prime q > p = prime(n) such that if d = q-p, then p, p+d, p+2d, ..., p+(p-1)d are all primes.

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

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A133278 Triangle read by rows, with n-th row the smallest non-constant n-term arithmetic progression of primes beginning with prime(n).

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Author

Keywords

Examples

Links

Crossrefs

Extensions