A033188 -id:A033188 - cp's OEIS frontend

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

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.

A343122 Consider the longest arithmetic progressions of primes from among the first n primes; a(n) is the smallest constant difference of these arithmetic progressions.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30

Offset: 2

Andres Cicuttin, Apr 05 2021

nonn

It seems that most terms are primorials (see comments in A338869 and A338238).

			For n=2, the first two primes are 2 and 3, the only subsequence of equidistant primes. The constant difference is 1, so a(2) = 1.
For n=3, there are three sequences of equidistant primes: {2,3} with constant difference 1, {3,5} with difference 2, and {2,5} with difference 3, so a(3) = 1 because 1 is the smallest constant difference among the three longest sequences.

Wikipedia, Primes in arithmetic progression

Cf. A338869, A338238, A002110 (Primorials), A343118, A033188.

nmax=100; (* Last n *)
maxlen=11 ; (* Maximum exploratory length of sequences of equidistant primes *)
(* a[n, p, s] returns the sequence of "s" equidistant primes with period "p" and last prime prime(n) if it exists, otherwise it returns {} *)
a[n_,period_,seqlen_]:=Module[{tab,test},
(* Building sequences of equidistant numbers ending with prime(n) *)
tab=Table[Prime[n]-k*period,{k,0,seqlen-1}];
(* Checking if all elements are primes and greater than 2 *)
test=(And@@PrimeQ@tab)&&(And@@Map[(#>2&),tab]);
Return[If[test,tab,{}]]];
atab={}; aterms={}; (* For every n, exploring all sequences of equidistant primes among the first n primes with n > 3 *)
Do[
Do[Do[
If[a[n,period,seqlen]!={},AppendTo[atab,{seqlen,period}]]
,{period,2,Ceiling[Prime[n]/(seqlen-1)],2}]
,{seqlen,2,maxlen}];
(* "longmax" is the length of the longest sequences *)
longmax=Sort[atab,#1[[1]]>#2[[1]]&][[1]][[1]];
(* Selecting the elements corresponding to the longest sequences *)
atab=Select[atab,#[[1]]==longmax&];
(* Saving the pairs {n, corresponding minimum periods} *)
AppendTo[aterms,{n,Min[Transpose[atab][[2]]]}]
,{n,4,nmax}];
(* Prepending the first two terms corresponding to the simple cases of first primes {2,3} and {2,3,5} *)
Join[{1,1},(Transpose[aterms][[2]])]

A113593 Minimal differences that appear in arithmetic progressions of primes.

Original entry on oeis.org

0, 1, 2, 6, 30, 150, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230

Offset: 1

Jon Wild, Jan 26 2006

nonn

Collapsed version of sequence A033188. If a conjecture about arithmetic prime progressions is correct (see A033188), this sequence is simply the primorial numbers with 150 inserted into the list. Zero is also included (since p,p is an arithmetic progression with difference 0).

			2 belongs in the sequence because the smallest d satisfying "n, n+d, n+2d are all prime" is 2. 4 does not belong, despite there existing triples {n, n+4, n+8} that are all prime--because a difference of 4 does not allow a new length of progression, i.e. there is no prime progression {n, n+4, n+8, n+12}.

Cf. A033188, A034386, A002110.

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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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A343122 Consider the longest arithmetic progressions of primes from among the first n primes; a(n) is the smallest constant difference of these arithmetic progressions.

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Mathematica

A113593 Minimal differences that appear in arithmetic progressions of primes.

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