A113872 -id:A113872 - 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.]
Thomas Bloom, Problem 200 and Problem 215, Erdős Problems.
Harvey Dubner and Harry 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.
Paul A. Pritchard, Andrew Moran and Anthony Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. 64 (1995), no. 211, 1337-1339.
Terence Tao, Erdős problem database, see no. 200, 215.
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

A033188 Minimal difference of any increasing arithmetic progression of n primes.

Original entry on oeis.org

0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690

Offset: 1

Eric W. Weisstein

The entries shown have been proved to be correct.

Jens Kruse Andersen, Status of minimal arithmetic progressions of primes
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Index entries for sequences related to primes in arithmetic progressions

Cf. A033189 (initial terms), A113872 (last terms).

Cf. A034386, A204189.

David W. Wilson conjectures that a(n) = n# (n primorial, A034386) for n >= 8. If this is true then a(22) onwards are: 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, ...

Entry revised Feb 02 2006 by N. J. A. Sloane, based on suggestions from Jens Kruse Andersen.

A033189 Smallest first term of arithmetic progression of n primes with difference A033188(n).

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 199, 199, 199, 60858179, 147692845283, 14933623, 834172298383, 894476585908771, 1275290173428391, 259268961766921, 1027994118833642281, 1424014323012131633, 1424014323012131633, 28112131522731197609

Offset: 1

Eric W. Weisstein

The entries shown have been proved to be correct.

J. K. Andersen, Primes in Arithmetic Progression Records.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Wikipedia, Prime Formula
Index entries for sequences related to primes in arithmetic progressions

Cf. A033188, A113872.

More terms from David W. Wilson
a(14) corrected by Gennady Gusev, Jul 07 2004
Oct 05 2004: Gennady Gusev reports that Jens Kruse Andersen found a(15) and that Gennady Gusev and Jens Kruse Andersen together found a(16) and a(17).
a(18) found by Gennady Gusev and Jens Kruse Andersen.
a(19)=a(20) found by Jaroslaw Wroblewski (jwr(AT)math.uni.wroc.pl), Dec 02 2008
a(21) found by Jaroslaw Wroblewski (jwr(AT)math.uni.wroc.pl), Dec 12 2008

A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

Original entry on oeis.org

2, 2, 3, 41, 5, 7, 7, 881, 3499, 199, 60858179, 147692845283, 14933623, 834172298383, 894476585908771, 1275290173428391, 259268961766921, 1027994118833642281

Offset: 1

Bernard Schott, Jun 05 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i.
The word "exactly" requires both i-d and i+n*d to be nonprime.
For the corresponding values of the last term, see A354744.
Without "exactly", we get A033189.
The primes in these arithmetic progressions need not be consecutive.
a(n) != A033189(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 881 and a(9) = 3499 found by Michael S. Branicky come from A354377.
a(19) > A033189(19) = 1424014323012131633 is not known, it is the smallest first term of an arithmetic progression of exactly 19 primes with a common difference d = 9699690; then a(20) = 1424014323012131633 and a(21) = 28112131522731197609.

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.

J. K. Andersen, Primes in Arithmetic Progression Records.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A033188, A033189, A113872, A354377, A354744.

A354744 Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).

Original entry on oeis.org

2, 3, 7, 59, 29, 157, 907, 2351, 5179, 2089, 60881279, 147692870693, 15293983, 834172688773, 894476586329191, 1275290173878841, 259268969935081, 1027994118842320951

Offset: 1

Bernard Schott, Jun 05 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime.
Without "exactly", we get A113872.
The primes in these arithmetic progressions need not be consecutive.
a(n) != 113872(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 2351 and a(9) = 5179, found by Michael S. Branicky come from A354376.
a(19) > A113872(19) = 1424014323186726053 is not known, it is the last term of the arithmetic progression of exactly 19 primes with a common difference d = 9699690 and first term = A354743(19); then a(20) = 1424014323196425743 and a(21) = 28112131522925191409.

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.

J. K. Andersen, Primes in Arithmetic Progression Records.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A033188, A033189, A113872, A354377, A354743.

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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A033188 Minimal difference of any increasing arithmetic progression of n primes.

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A033189 Smallest first term of arithmetic progression of n primes with difference A033188(n).

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A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

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A354744 Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).

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