A260751 -id:A260751 - cp's OEIS frontend

A204189 Benoît Perichon's 26 primes in arithmetic progression.

43142746595714191, 48425980631694091, 53709214667673991, 58992448703653891, 64275682739633791, 69558916775613691, 74842150811593591, 80125384847573491, 85408618883553391, 90691852919533291, 95975086955513191, 101258320991493091, 106541555027472991, 111824789063452891, 117108023099432791, 122391257135412691, 127674491171392591, 132957725207372491, 138240959243352391, 143524193279332291, 148807427315312191, 154090661351292091, 159373895387271991, 164657129423251891, 169940363459231791, 175223597495211691

Offset: 1

Jonathan Sondow, Jan 14 2012

Longest known arithmetic progression of primes as of Jan 14, 2012.

Discovered on Apr 12 2010 by Benoît Perichon using software by Jaroslaw Wroblewski and Geoff Reynolds in a distributed PrimeGrid project.

R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, A5 and A6.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1989, p. 224.

J. K. Andersen, Primes in Arithmetic Progression Records.
T. Eisner and R. Nagel, Arithmetic progressions-an operator theoretic view, Discrete and continuous dynamical systems series S, Volume 6, Number 3, June 2013 pp. 657-667; doi:10.3934/dcdss.2013.6.657. - From _N. J. A. Sloane_, Feb 03 2013
A. Granville, Prime Number Patterns, Amer. Math. Monthly 115 (2008), 279-296.
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), 481-547.
PrimeGrid, AP26 Search.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Wikipedia, Primes in arithmetic progression.
J. Wroblewski, How to search for 26 primes in arithmetic progression?, May 23, 2008.
Index entries for sequences related to primes in arithmetic progressions

Cf. A002110, A033188, A033189, A033290, A260751, A261140, A327760, A363980, A374949.

a[1] := 43142746595714191; a[n_] := a[n] = a[n - 1] + 5283234035979900; Table[a[n], {n, 26}] (* Alonso del Arte, Jan 14 2012 *)
Range[ 43142746595714191, 175223597495211691, 5283234035979900] (* Michael Somos, Jan 15 2012 *)

a(n)=5283234035979900*n+37859512559734291 \\ Charles R Greathouse IV, Jan 15 2012

a(n) = 43142746595714191 + 5283234035979900*(n-1) for n = 1, 2, ..., 26.

a(n) = 43142746595714191 + 23681770*23#*(n-1) for n = 1..26, where 23# = 2*3*5*7*11*13*17*19*23 = 223092870 = A002110(9).

A261140 a(n) = 3486107472997423 + (n-1)*371891575525470.

Original entry on oeis.org

3486107472997423, 3857999048522893, 4229890624048363, 4601782199573833, 4973673775099303, 5345565350624773, 5717456926150243, 6089348501675713, 6461240077201183, 6833131652726653, 7205023228252123, 7576914803777593, 7948806379303063, 8320697954828533

Offset: 1

Marco Ripà, Aug 10 2015

The terms n = 1..26 are prime. This is the longest sequence of primes in arithmetic progression with smallest end, a(26)=12783396861134173, known as of August 10, 2015.

			a(26) = 3486107472997423 + 25*371891575525470 = 12783396861134173 is prime.

Colin Barker, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002110, A033188, A033290, A204189, A260751, A327760, A363980, A374949.

[3486107472997423+(n-1)*371891575525470: n in [1..20]];

Table[3486107472997423 + (n - 1) 371891575525470, {n, 1, 20}]
LinearRecurrence[{2,-1},{3486107472997423,3857999048522893},20] (* Harvey P. Dale, May 14 2022 *)

Vec(-x*(3114215897471953*x-3486107472997423)/(x-1)^2 + O(x^40)) \\ Colin Barker, Aug 25 2015

a(n) = 3486107472997423 + (n-1)*1666981*A002110(9).

G.f.: -x*(3114215897471953*x-3486107472997423) / (x-1)^2. - Colin Barker, Aug 25 2015

A033290 Ten consecutive primes in arithmetic progression.

Original entry on oeis.org

100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229719, 100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229929, 100996972469714247637786655587969840329509324689190041803603417758904341703348882159067230139

Offset: 0

Manfred Toplic

This was the first known case, found in 1998. The full 10 terms are linked below. - Jens Kruse Andersen, Jun 30 2014

Jens Kruse Andersen, Table of n, a(n) for n = 0..9
J. K. Andersen, The largest known CPAP-10
H. Dubner et al., Ten consecutive primes in arithmetic progression
T. Forbes, Ten consecutive primes in arithmetic progression
Tanya Khovanova, Non Recursions
Manfred Toplic, Nine and ten primes project
Eric Weisstein's World of Mathematics, Primes in Arithmetic Progression
Index entries for sequences related to primes in arithmetic progressions

Cf. A033188, A204189, A260751, A261140, A327760, A363980, A374949.

N*m + x + 210*b, b = 0..9.

a(n) = a(0)+210*n, and a(n+1) = nextprime(a(n)+1). - Jens Kruse Andersen, Jun 30 2014

A327760 Primes in Rob Gahan's arithmetic progression of 27 primes.

Original entry on oeis.org

224584605939537911, 242720302537486841, 260855999135435771, 278991695733384701, 297127392331333631, 315263088929282561, 333398785527231491, 351534482125180421, 369670178723129351, 387805875321078281, 405941571919027211, 424077268516976141, 442212965114925071

Offset: 1

Felix Fröhlich, Sep 25 2019

This arithmetic progression of 27 primes (AP27) was discovered by Rob Gahan on 23 September 2019 as part of PrimeGrid's AP27 Search subproject (cf. Goetz, 2019).

Felix Fröhlich, Table of n, a(n) for n = 1..27
M. Goetz, World's First AP27!!!, PrimeGrid forum, Sep 23, 2019.
PrimeGrid, 224584605939537911+81292139*23#*n for n=0..26
PrimeGrid, Official announcement of the AP27
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 26.
Index entries for sequences related to primes in arithmetic progressions

Cf. A033188, A033290, A204189, A260751, A261140, A363980, A374949.

A327760[n_] := 224584605939537911 + (n-1)*18135696597948930;
Array[A327760, 27] (* Paolo Xausa, Jan 30 2024 *)

vector(27, t, 224584605939537911+81292139*223092870*(t-1))

A363980 Tom Greer's arithmetic progression of 27 primes.

Original entry on oeis.org

277699295941594831, 315809464967513821, 353919633993432811, 392029803019351801, 430139972045270791, 468250141071189781, 506360310097108771, 544470479123027761, 582580648148946751, 620690817174865741, 658800986200784731, 696911155226703721, 735021324252622711

Offset: 1

Marco Ripà, Jun 30 2023

At the time of submission (June 2023), this sequence is the arithmetic progression of 27 primes having the largest known initial and final term and it was found by Tom Greer on 26 May 2023 as part of PrimeGrid's AP27, running the program AP26 (this is the second known AP27 to date, see A327760).

			a(3) = 277699295941594831 + 2*170826477*223092870 is prime.

Marco Ripà, Table of n, a(n) for n = 1..27
PrimeGrid, Official announcement of the second known AP27
PrimeGrid forum, Message 163220 (World's Second AP27!!!)

Cf. A033188, A033290, A204189, A260751, A261140, A327760, A374949.

A363980[n_]:=277699295941594831 + (n-1)*38110169025918990;
Array[A363980, 27] (* Paolo Xausa, Jan 30 2024 *)

vector(27, t, 277699295941594831+170826477*223092870*(t-1))

a(n+1) = 277699295941594831 + n*170826477*223092870, for n = 0, 1, ..., 26.

A374949 Michael Kwok's arithmetic progression of 27 primes.

Original entry on oeis.org

605185576317848261, 639847242910261121, 674508909502673981, 709170576095086841, 743832242687499701, 778493909279912561, 813155575872325421, 847817242464738281, 882478909057151141, 917140575649564001, 951802242241976861, 986463908834389721, 1021125575426802581

Offset: 1

Marco Ripà, Jul 24 2024

At the time of submission (July 2024), this sequence is the arithmetic progression of 27 primes having the largest known initial and final term and it was found by Michael Kwok on 10 December 2023 as part of the project PrimeGrid, running the program AP26 (this is the third known AP27 to date, see A327760 and A363980).

			a(3) = 605185576317848261 + 2*34661666592412860 is prime.

PrimeGrid forum, Message 167007 (World's Third AP27!!!).

Cf. A033188, A033290, A204189, A260751, A261140, A327760, A363980.

A374949[n_]:=605185576317848261 + (n-1)* 34661666592412860; Array[A374949, 27]

vector(27, t, 605185576317848261+155368778*223092870*(t-1))

a(n+1) = 605185576317848261 + n*34661666592412860, for n = 0, 1, ..., 26.

A317163 a(n) = 48277590120607451 + (n-1)*8440735245322380.

Original entry on oeis.org

48277590120607451, 56718325365929831, 65159060611252211, 73599795856574591, 82040531101896971, 90481266347219351, 98922001592541731, 107362736837864111, 115803472083186491, 124244207328508871, 132684942573831251, 141125677819153631, 149566413064476011

Offset: 1

Marco Ripà, Jul 23 2018

a(1), a(2), ..., a(26) are prime. As of Jul 23 2018, this is one of the longest known sequences of primes in arithmetic progression, and was found by Bruce E. Slade in 2017.

			a(26) = 48277590120607451 + 25*37835074*223092870 = 259295971253666951 is prime.

Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.

Cf. A002110, A204189, A260751, A261140.

List([1..26],n->55837783597462913+(n-1)*13858932213216090); # Marco Ripà, Aug 10 2018

seq(48277590120607451+(n-1)*8440735245322380,n=1..26); # Marco Ripà, Aug 10 2018

Table[48277590120607451 + (n - 1) 8440735245322380, {n, 1, 26}]

a(n) = 48277590120607451 + a(n-1)*37835074*23#, where 23# := 2*3*5*7*11*13*17*19*23 = 223092870.

A317164 a(n) = 55837783597462913 + (n-1)*13858932213216090.

Original entry on oeis.org

55837783597462913, 69696715810679003, 83555648023895093, 97414580237111183, 111273512450327273, 125132444663543363, 138991376876759453, 152850309089975543, 166709241303191633, 180568173516407723, 194427105729623813, 208286037942839903, 222144970156055993

Offset: 1

Marco Ripà, Jul 23 2018

The terms for n = 1..26 are prime. As of Jul 23 2018, this is one of the longest known sequences of primes in arithmetic progression.

			a(26) = 55837783597462913 + 25*62121807*223092870 = 402311088927865163 is prime.

Jens Kruse Andersen, All known AP24 to AP26.
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), 481-547.
PrimeGrid, AP26 Search.
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Wikipedia, Primes in arithmetic progression.

Cf. A002110, A204189, A260751, A261140, A317163.

List([1..25],n->55837783597462913+(n-1)*13858932213216090); # Muniru A Asiru, Jul 24 2018

seq(55837783597462913+(n-1)*13858932213216090,n=1..15); # Muniru A Asiru, Jul 24 2018

Table[55837783597462913 + (n - 1) 13858932213216090, {n, 1, 25}]

a(n) = 455837783597462913 + a(n-1)*62121807*23#, where 23# := 2*3*5*7*11*13*17*19*23 = 223092870.

A260939 Thirteen primes in arithmetic progression with difference 60060 and minimal initial term.

Original entry on oeis.org

4943, 65003, 125063, 185123, 245183, 305243, 365303, 425363, 485423, 545483, 605543, 665603, 725663

Offset: 1

Marco Ripà, Aug 05 2015

This sequence is 13 primes long and was discovered by W. N. Seredinsky.

			a(4) = 4943 + 3*60060 = 185123.

Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Primes in arithmetic progression.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002110, A033290, A113827, A260751.

[4943+(n-1)*60060: n in [1..13]]; // Bruno Berselli, Aug 10 2015

Table[4943 + (n - 1) 60060, {n, 1, 13}] (* Bruno Berselli, Aug 10 2015 *)

a(n)=60060*n-55117 \\ Charles R Greathouse IV, Aug 25 2017

[4943+(n-1)*60060 for n in (1..13)] # Bruno Berselli, Aug 10 2015

a(n) = 4943 + (n-1)*60060 = 4943 + (n-1)*2*A002110(6).

A261152 a(n) = 161004359399459161 + (n-1)*10644900609172830.

Original entry on oeis.org

161004359399459161, 171649260008631991, 182294160617804821, 192939061226977651, 203583961836150481, 214228862445323311, 224873763054496141, 235518663663668971, 246163564272841801, 256808464882014631, 267453365491187461, 278098266100360291, 288743166709533121

Offset: 1

Marco Ripà, Aug 10 2015

The terms n = 1..26 are prime. This is the longest and largest sequence of primes in arithmetic progression, a(26)=427126874628779911, known as of August 10, 2015.

			a(26) = 161004359399459161 + 25*10644900609172830 = 427126874628779911 is prime.

Jens Kruse Andersen, All known AP24 to AP26.
Wikipedia, Largest known primes in AP.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002110, A204189, A260751.

[161004359399459161+(n-1)*10644900609172830: n in [1..20]]; // Bruno Berselli, Aug 23 2015

Table[161004359399459161 + (n - 1) 10644900609172830, {n, 1, 20}] (* Bruno Berselli, Aug 23 2015 *)

a(n) = 161004359399459161 + (n-1)*47715109*A002110(9).

G.f.: x*(161004359399459161 - 150359458790286331*x)/(1 - x)^2. [Bruno Berselli, Aug 23 2015]

cp's OEIS Frontend

A204189 Benoît Perichon's 26 primes in arithmetic progression.

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A261140 a(n) = 3486107472997423 + (n-1)*371891575525470.

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A033290 Ten consecutive primes in arithmetic progression.

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A327760 Primes in Rob Gahan's arithmetic progression of 27 primes.

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A363980 Tom Greer's arithmetic progression of 27 primes.

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A374949 Michael Kwok's arithmetic progression of 27 primes.

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A317163 a(n) = 48277590120607451 + (n-1)*8440735245322380.

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