A133277 -id:A133277 - 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

A086786 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same prime signature.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 481, 485, 489, 493, 497, 501, 505, 635, 707, 779, 851, 923, 995, 1067, 1139, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 3841

Offset: 1

Amarnath Murthy, Sep 02 2003

In this sequence "smallest" means that the last term of the arithmetic progression is minimized. A113460 minimizes the first term.

			Triangle begins:
1
2,3
3,5,7
5,11,17,23
5,11,17,23,29
7,37,67,97,127,157
481,485,489,493,497,501,505
635,707,779,851,923,995,1067,1139
635,707,779,851,923,995,1067,1139,1211
199,409,619,829,1039,1249,1459,1669,1879,2089
...

Index entries for sequences related to primes in arithmetic progressions

A087310 contains the corresponding common differences, A087308 the initial terms, A087309 the final terms.
Cf. A087308, A087309, A087310, A114660, A114663, A114670.
Row 10 is the same for A113470, A133276, A133277 and listed as A033168.

Edited and extended by David Wasserman, Jan 08 2006

Further edits by N. J. A. Sloane, Oct 17 2007

A113470 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same number of divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 35, 65, 95, 125, 155, 185, 215, 635, 707, 779, 851, 923, 995, 1067, 1139, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 3841, 3973

Offset: 1

David Wasserman, Jan 08 2006

In this sequence "smallest" means that the last term of the arithmetic progression is minimized and if there is still a choice then we minimize the common difference of the arithmetic progression.

			From _M. F. Hasler_, Jan 02 2020: (Start)
The triangle starts
    n | row n
   ---+------------
    1 |    1,
    2 |    2,    3,
    3 |    3,    5,    7,
    4 |    5,   11,   17,   23,
    5 |    5,   11,   17,   23,   29,
    6 |    7,   37,   67,   97,  127,  157,
    7 |   35,   65,   95,  125,  155,  185,  215,
    8 |  635,  707,  779,  851,  923,  995, 1067, 1139,
    9 |  635,  707,  779,  851,  923,  995, 1067, 1139, 1211,
   10 |  199,  409,  619,  829, 1039, 1249, 1459, 1669, 1879, 2089,
   11 | 3841, 3973,  ...
Most rows so far consist of primes with 2 divisors, rows 7, 8, 9 and 11 have squarefree semiprimes with 4 divisors.
Row 10 is A033168; also row 10 of A086786, A133276 and A133277. (End)

OEIS wiki, Primes in arithmetic progression.

Cf. A086786, A090547, A090548, A090549.

T(n,k) = A090547(n) + (k-1)*A090549(n). - R. J. Mathar, May 11 2007

A133276 Triangle read by rows: row n gives the first arithmetic progression of n primes with minimal distance, cf. A033188.

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, 60858179, 60860489, 60862799, 60865109, 60867419, 60869729, 60872039, 60874349, 60876659, 60878969, 60881279

Offset: 1

N. J. A. Sloane, Oct 17 2007

The first 10 rows (i.e., 55 terms) are the same as for A133277 (where the final term is minimal), but here a(56) = T(11,1) = 608581797 while A133277(11,1) = 110437. - M. F. Hasler, Jan 02 2020

			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
  ...
Row 10 is the same as in A086786, A113470, A133277, and listed as A033168. - _M. F. Hasler_, Jan 02 2020

For common differences see A033188, for initial terms see A033189.
Different from A133277 (from T(11,1) = a(56) on).
Cf. A086786, A113470, A033168.

AP:=proc(i,d,l) [seq(i + (j-1)*d, j=1..l )]; end;

A279021 Triangle read by rows, giving the arithmetic progressions of prime-indexed primes in A278735.

Original entry on oeis.org

3, 3, 5, 5, 11, 17, 353, 431, 509, 587, 13297, 21937, 30577, 39217, 47857, 1561423, 2716423, 3871423, 5026423, 6181423, 7336423, 291461857, 373881397, 456300937, 538720477, 621140017, 703559557, 785979097

Offset: 1

Bobby Jacobs, Dec 03 2016

Suggested by Charles R Greathouse IV in A278735.
The first set of 4 prime-indexed primes in arithmetic progression (353, 431, 509, and 587) contains consecutive terms of A142160.
The first set of 5 prime-indexed primes in arithmetic progression contains 3 numbers that are anagrams of each other (13297, 21937, and 39217).

			a(7) = 353, a(8) = 431, a(9) = 509, and a(10) = 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.
The triangle begins:
3;
3, 5;
5, 11, 17;
353, 431, 509, 587;
13297, 21937, 30577, 39217, 47857;
1561423, 2716423, 3871423, 5026423, 6181423, 7336423;
...

Cf. A006450, A133277, A142160, A274825, A278735, A279062.

a(22)-a(28) from Charles R Greathouse IV, Dec 27 2016

A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

Original entry on oeis.org

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 0

Manuel Valdivia, Apr 22 1998

Since 210 == 1 (mod 11), a progression of primes with difference 210 can't have more than ten terms because there is exactly one multiple of 11 within each run of eleven consecutive terms. For example, 2089 + 210 = 2299 = 11^2 * 19. - Alonso del Arte, Dec 22 2017, edited by M. F. Hasler, Jan 02 2020

After 199, the next prime which starts a CPAP-10 with common gap 210 is 243051733. See A094220 for further starting points. - M. F. Hasler, Jan 02 2020

Paul Glendinning, Math in Minutes: 200 Key Concepts Explained in an Instant. New York, London: Quercus (2013): pp. 316-317.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 143.

Tanya Khovanova, Non Recursions.
OEIS wiki, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Row 10 of A086786, A113470, A133276, A133277.

Cf. A094220.

199 + 210*Range[0, 9] (* Paolo Xausa, Sep 14 2024 *)

forprime(p=1,,for(i=1,9,isprime(p+i*210)||next(2)); return([p+d|d<-[0..9]*210])) \\ M. F. Hasler, Jan 02 2020

a(n) = a(0) + n*210 for 0 <= n <= 9. - M. F. Hasler, Jan 02 2020

A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 7, 19, 31, 43, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 881, 1091, 1301, 1511, 1721, 1931, 2141, 2351, 3499, 3709, 3919, 4129, 4339, 4549, 4759, 4969, 5179, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 1

Bernard Schott, May 29 2022

For the corresponding values of the first term, the last term and the common difference of these arithmetic progressions, see respectively A354377, A354376 and A354484.
Without "exactly", we get A133277.
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 progressions is A006560(n).)

			Triangle begins:
    2;
    2,    3;
    3,    5,    7;
    7,   19,   31,   43;
    5,   11,   17,   23,   29;
    7,   37,   67,   97,  127,  157;
    7,  157,  307,  457,  607,  757,  907;
  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.

Michael S. Branicky, Table of n, a(n) for n = 1..231 (using A354376 and A354377)

Cf. A006560, A133277, A354376, A354377, A354484.

T(n, 1) = A354377.

T(n, n) = A354376.

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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A086786 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same prime signature.

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A113470 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same number of divisors.

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A133276 Triangle read by rows: row n gives the first arithmetic progression of n primes with minimal distance, cf. A033188.

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A279021 Triangle read by rows, giving the arithmetic progressions of prime-indexed primes in A278735.

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A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

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A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

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