A113827 -id:A113827 - cp's OEIS frontend

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

N. J. A. Sloane

In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.

			n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.

R. K. Guy, Unsolved Problems in Number Theory, A5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Ben Green and Terence Tao, A bound for progressions of length k in the primes
Andrew Granville, Prime number patterns, American Mathematical Monthly 115 (2008), pp. 279-296.
A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp.64 (1995), no.211, 1337-1339.
Index entries for sequences related to primes in arithmetic progressions

For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.

Cf. A006560, A096003, A113830-A113834, A088430.

(* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]

Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - Charles R Greathouse IV, Feb 26 2013

a(11)-a(13) from Michael Somos, Mar 14 2004
a(14) and corrected version of a(7) from Hugo Pfoertner, Apr 27 2004
a(15)-a(17) from Don Reble, Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26 2006
Entry revised by N. J. A. Sloane, Jan 26 2006, Oct 17 2007

A115608 Number of triples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

1, 46, 1500, 59504, 2856331, 157300309, 9565120490

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The only 3-tuple less than 10 is {3,5,7}.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115609-A115613.

A115613 Number of 8-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 8, 243, 5222, 140667

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The least 8-tuple is {199, 409, 619, 829, 1039, 1249, 1459, 1669}, which is also the beginning of the least 10-tuple.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115608-A115612.

A113832 Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the pairwise averages are all primes, having the smallest largest element.

Original entry on oeis.org

3, 7, 3, 7, 19, 3, 11, 23, 71, 5, 29, 53, 89, 113, 3, 11, 83, 131, 251, 383, 5, 29, 113, 269, 353, 449, 509, 5, 17, 41, 101, 257, 521, 761, 881, 23, 431, 503, 683, 863, 1091, 1523, 1871, 2963, 31, 1123, 1471, 1723, 3463, 3571, 4651, 5563, 5743, 6991

Offset: 2

N. J. A. Sloane, Jan 25 2006

If there is more than one set with the same smallest last element, choose the lexicographically earliest solution.

For distinct primes, the solution for n=5 is {5, 29, 53, 89, 173}.

			Triangle begins:
3, 7
3, 7, 19
3, 11, 23, 71
5, 29, 53, 89, 113
3, 11, 83, 131, 251, 383
5, 29, 113, 269, 353, 449, 509
The set of primes generated by {5, 29, 53, 89, 113} is {17, 29, 41, 47, 59, 59, 71, 71, 83, 101}.

Antal Balog, The prime k-tuplets conjecture on average, in "Analytic Number Theory" (eds. B. C. Berndt et al.) Birkhäuser, Boston, 1990, pp. 165-204. [Background]

Toshitaka Suzuki, Table of n, a(n) for n = 2..91
Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Andrew Granville, Prime number patterns

Cf. A113827-A113831, A113833, A113834, A088430.

See A115631 for the case when all pairwise averages are distinct primes.

More terms from T. D. Noe, Feb 01 2006

A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037

Offset: 1

N. J. A. Sloane, Oct 17 2007

The first 10 rows (i.e., 55 terms) are the same as for A133276 (where the common distance is minimal), but here T(11,1) = a(56) = 110437 while A133276(11,1) = 60858179. - M. F. Hasler, Jan 02 2020

For any prime p there is a p-AP (arithmetic progression of p primes) starting with p, where the common distance is given by A088430. For n between prime(k-1) and prime(k), there may be an n-AP starting at prime(k) (but not earlier) with a smaller common distance, given in A061558. - M. F. Hasler, Sep 17 2024

			Triangle begins:
    2;
    2,   3;
    3,   5,   7;
    5,  11,  17,  23;
    5,  11,  17,  23,   29;
    7,  37,  67,  97,  127,  157;
    7, 157, 307, 457,  607,  757,  907;
  199, 409, 619, 829, 1039, 1249, 1459, 1669;
  199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879;
  199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089;
  ...

For common differences, see A093364. For initial terms, see A113827. For final terms, see A005115.
Differs from A133276 (from T(11,1) = a(56) on).
See also A061558 (distance in earliest n-AP), A088430 (same for primes), A231017 (second term in p-AP starting with p), A061558 (distance of n-AP starting at the smallest possible prime).

A-numbers in the Name and Crossrefs sections corrected by Bobby Jacobs, Dec 10 2016

Name edited by M. F. Hasler, Jan 02 2020

A115609 Number of 4-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 11, 318, 10257, 389606, 17636879, 909031090

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The least 4-tuple is {5,11,17,23}.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115608-A115613.

A115612 Number of 7-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 1, 32, 959, 25018, 793022

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The least 7-tuple is {7,157,307,457,607,757,907}.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115608-A115613.

A115610 Number of 5-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 2, 58, 1283, 39866, 1495343, 65348949

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The least 5-tuple is {5,11,17,23,29}.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115608-A115613.

A115611 Number of 6-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 14, 263, 7304, 227456, 8433923

Offset: 1

T. D. Noe, Jan 25 2006

nonn

			The least 6-tuple is {7,37,67,97,127,157}.

See A113827

Cf. A113827 (prime beginning minimal n-tuple of primes in AP), A115608-A115613.

A354377 Initial terms associated with the arithmetic progressions of primes of A354376.

Original entry on oeis.org

2, 2, 3, 7, 5, 7, 7, 881, 3499, 199, 75307, 110437, 4943, 31385539, 115453391, 53297929, 3430751869, 4808316343, 8297644387, 214861583621, 5749146449311

Offset: 1

Bernard Schott, May 26 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) = first term i.
The adverb "exactly" requires both i-d and i+n*d to be nonprime (see A113827).
For the corresponding values of the last term, see A354376.
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) != A113827(n) for n = 4, 8, 9, 11. - Michael S. Branicky, May 26 2022

			The first few corresponding arithmetic progressions are:
n = 1 (2);
n = 2 (2, 3);
n = 3 (3, 5, 7);
n = 4 (7, 19, 31, 43);
n = 5 (5, 11, 17, 23, 29);
n = 6 (7, 37, 67, 97, 127, 157);
n = 7 (7, 157, 307, 457, 607, 757, 907)...

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

Cf. A006560, A113827, A354376.

a(8)-a(21) from Michael S. Branicky, May 26 2022

cp's OEIS Frontend

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

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A115608 Number of triples of primes in arithmetic progression less than 10^n.

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A115613 Number of 8-tuples of primes in arithmetic progression less than 10^n.

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A113832 Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the pairwise averages are all primes, having the smallest largest element.

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A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

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A115609 Number of 4-tuples of primes in arithmetic progression less than 10^n.

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A115612 Number of 7-tuples of primes in arithmetic progression less than 10^n.

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A115610 Number of 5-tuples of primes in arithmetic progression less than 10^n.

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A115611 Number of 6-tuples of primes in arithmetic progression less than 10^n.

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A354377 Initial terms associated with the arithmetic progressions of primes of A354376.

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