A005115 -id:A005115 - 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

A273919 Number of 9-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 3, 57, 984, 22551

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 9-tuple is {199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879}, which is also the beginning of the least 10-tuple. This explains a(n)=0 for n<4.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A115608, A115609, A115610, A115611, A115612, A115613, A273920, A273921, A273922, A273923.

A273920 Number of 10-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 1, 5, 145, 2969

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 10-tuple is {199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089}, which explains a(4)=1 (and a(n)=0 for n<4).

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273921, A273922, A273923.

A273921 Number of 11-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 253, 5561

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 11-tuple is {110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037} (this is also the beginning of the least 12-tuple). This is one of the 15 11-tuples corresponding to a(6)=15.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273922, A273923.

A273922 Number of 12-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 42, 715

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 12-tuple is {110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037, 262897}.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273921, A273923.

A273923 Number of 13-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 52

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 13-tuple is {4943, 65003, 125063, 185123, 245183, 305243, 365303, 425363, 485423, 545483, 605543, 665603, 725663}.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273921, A273922.

A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

Original entry on oeis.org

2, 3, 7, 43, 29, 157, 907, 2351, 5179, 2089, 375607, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

Bernard Schott, May 24 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A005115.
For the corresponding values of the first term, and the common difference, see A354377 and A354484. For the actual arithmetic progressions, see A354485.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)
a(n) != A005115(n), because A005115(n) + A093364(n) is prime for n = 4, 8, 9, 11. - Michael S. Branicky, May 24 2022

			The arithmetic progression (5, 11, 17, 23) with common difference 6 contains 4 primes, but 29 = 23+6 is also prime, hence a(4) != 23.
The arithmetic progression (7, 19, 31, 43) with common difference 12 also contains 4 primes, and 7-12 < 0 and 43+12 = 55 is composite; moreover this arithmetic progression is the smallest such progression with exactly 4 primes, hence a(4) = 43.

R. K. Guy, Unsolved Problems in Number Theory, A5, Arithmetic progressions of primes.

Cf. A005115, A006560, A093364, A354377, A354484, A354485.

from sympy import isprime, nextprime
def a(n):
    if n < 3: return [2, 3][n-1]
    p = 2
    while True:
        for d in range(2, (p-3)//(n-1)+1, 2):
            if isprime(p+d) or isprime(p-n*d): continue
            if all(isprime(p-j*d) for j in range(1, n)): return p
        p = nextprime(p)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, May 24 2022

a(4) corrected and a(8)-a(13) from Michael S. Branicky, May 24 2022

a(14)-a(21) derived using A005115 and A093364 by Michael S. Branicky, May 24 2022

A373888 a(n) is the length of the longest arithmetic progression of primes ending with prime(n).

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 3, 3, 4, 5, 3, 2, 4, 4, 3, 5, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 5, 5, 3, 4, 4, 4, 6, 4, 4, 5, 3, 4, 4, 4, 5, 4, 3, 4, 5, 4, 4, 4, 4, 5, 6, 4, 4, 5, 3, 4, 5, 5, 4, 6, 4, 4, 4, 3, 4, 4, 6, 4, 4, 5, 3, 4, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 4, 4, 6, 4, 5, 4, 4, 3, 4, 6, 5, 4

Offset: 1

Robert Israel, Aug 11 2024

nonn

a(n) is the greatest k such that there exists d > 0 such that A000040(n) - j*d is prime for j = 0 .. k-1.
The first appearance of m in this sequence is at A000720(A005115(m)).
Conjectures: a(n) >= 3 for n >= 13.
Limit_{n -> oo} a(n) = oo.

			a(4) = 3 because the 4th prime is 7 and there is an arithmetic progression of 3 primes ending in 7, namely 3, 5, 7, and no such arithmetic progression of 4 primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A005115, A373887.

f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= ithprime(n);
  for i from n-1 to 1 by -1 do
    d:= s - ithprime(i);
    if s - m*d < 2 then return m fi;
    for j from 2 while isprime(s-j*d) do od;
    m:= max(m, j);
  od;
  m
end proc:
map(f, [$1..100]);

A278735 Smallest prime-indexed prime ending an arithmetic progression of n prime-indexed primes.

Original entry on oeis.org

3, 5, 17, 587, 47857, 7336423, 785979097

Offset: 1

Bobby Jacobs, Nov 27 2016

The first set of 4 prime-indexed primes in arithmetic progression (353, 431, 509, and 587) contains consecutive terms of A142160.

			a(4) = 587 because 353 = prime(prime(20)), 431 = prime(prime(23)), 509 = prime(prime(25)), 587 = prime(prime(28)), and 431-353 = 509-431 = 587-509 = 78.
From _Charles R Greathouse IV_, Nov 27 2016: (Start)
The corresponding arithmetic progressions are
3;
3, 5;
5, 11, 17;
353, 431, 509, 587;
13297, 21937, 30577, 39217, 47857;
1561423, 2716423, 3871423, 5026423, 6181423, 7336423;
and with the main diagonal being terms of this sequence. (End)

Right border of A279021.

Cf. A005115, A006450, A142160, A274825, A279062.

findAP(len)=my(t); if(len<3, return(v[len])); for(i=len, #v, for(j=1, i-len+1, t=(v[i]-v[j])/(len-1); if(denominator(t)>1, next); forstep(k=v[j]+t, v[i]-t, t, if(!setsearch(v, k), next(2))); return(vector(len, k, v[j]+(k-1)*t)))); "not found"
listPIP(lim)=my(v=List(), p); forprime(q=2, lim, if(isprime(p++), listput(v, q))); Vec(v)
v=listPIP(1e7);
apply(findAP, [1..6]) \\ Charles R Greathouse IV, Nov 27 2016

a(7) from Charles R Greathouse IV, Dec 27 2016

A113459 Least number that begins an arithmetic progression of n numbers with the same prime signature.

Original entry on oeis.org

1, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13

Offset: 1

David Wasserman, Jan 08 2006

Initial terms of arithmetic progressions described in A113460. - N. J. A. Sloane, Oct 18 2007
Conjecture: For n > 1, a(n) = A007918(n). - David Wasserman, Jan 08 2006
I disagree with that conjecture! Ignoring the initial terms, this will agree with A007918 up to some point and then (presumably) drop below A007918. The initial term in the arithmetic progression (of length n) must be >= n, but it is likely to be less than A007918(n) if n is large. - N. J. A. Sloane, Oct 18 2007

Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A007918, A087309, A113460.

Cf. A113461, A127781, A007917, A061558.

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar.

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.

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A273919 Number of 9-tuples of primes in arithmetic progression less than 10^n.

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A273920 Number of 10-tuples of primes in arithmetic progression less than 10^n.

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A273921 Number of 11-tuples of primes in arithmetic progression less than 10^n.

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A273922 Number of 12-tuples of primes in arithmetic progression less than 10^n.

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A273923 Number of 13-tuples of primes in arithmetic progression less than 10^n.

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A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

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A373888 a(n) is the length of the longest arithmetic progression of primes ending with prime(n).

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Maple

A278735 Smallest prime-indexed prime ending an arithmetic progression of n prime-indexed primes.

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