A052239 -id:A052239 - cp's OEIS frontend

A089236 Duplicate of A052239.

Original entry on oeis.org

251, 111497, 74453, 1397609, 642427, 5321191, 23921299, 55410683

Offset: 1

dead

A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

Original entry on oeis.org

2, 2, 3, 251, 9843019, 121174811

Offset: 1

N. J. A. Sloane

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004
From Daniel Forgues, Jan 17 2011: (Start)
It is conjectured that there are arithmetic progressions of n consecutive primes for any n.
Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)
a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022

			First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, The smallest known CPAP-k.
Thomas Bloom, Let k >= 3. Are there k consecutive primes in arithmetic progression?, Erdős Problems.
Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
Terence Tao, Erdős problem database, see no. 141.
Manfred Toplic, The nine and ten primes project, 2004.
Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A006562, A093364, A126989.
a(5) corresponds to A052243(20) followed by A052243(21) 9843049.
Cf. A089180: indices primes a(n).
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n.
Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - M. F. Hasler, Oct 27 2018

Edited by Daniel Forgues, Jan 17 2011

A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

Original entry on oeis.org

9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

Each set has a constant difference of 30, for all of the terms listed so far.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.
The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018
About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.

Zak Seidov, Table of n, a(n) for n = 1..241 (all terms up to 3*10^9)
Jens K. Andersen, The Largest Known CPAP's, updated Sept. 2018
OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A052239: start of first CPAP-4 with common difference 6n.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Cf. A006560: first prime to start a CPAP-n.

Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)

A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018

Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.

A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - M. F. Hasler, Oct 27 2018, edited Jan 02 2020.

a(16)-a(22) from Donovan Johnson, Sep 05 2008
Reference added by Harvey P. Dale, Jun 22 2013
Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018

A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

Original entry on oeis.org

121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.
All terms are congruent to 9 (mod 14). - Zak Seidov, May 03 2017
The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3*10^11, cf. A210727. [With a slope of a(n)/n ~ 5*10^8 this would correspond to n ~ 600.] This sequence consists of first members of pairs of consecutive primes in A059044. Conversely, a pair of consecutive primes in this sequence starts a CPAP-7. This must have a common difference >= 210. As of today, the smallest known CPAP-7 starts at 382003672700092872707633 ~ 3.8*10^23, cf. Andersen link. - M. F. Hasler, Oct 27 2018
The common difference of 60 first occurs at a larger-than-expected prime. The first CPAP-6 with common difference 90 starts at 8560443932347. The first CPAP-6 with common difference 120 starts at 1925601119017087. - Jerry M Lagrou, Jan 01 2024

Zak Seidov, Table of n, a(n) for n = 1..102
Jens K. Andersen, The Largest Known CPAP's, updated Sep 2018.
OEIS Wiki, Consecutive primes in arithmetic progression, updated Jan 2020.
Index entries for sequences related to primes in arithmetic progressions

Cf. A006560: first prime to start a CPAP-n.
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A052239: starting prime of first CPAP-4 with common difference 6n.
Cf. A059044: starting primes of CPAP-5.
Cf. A210727: starting primes of CPAP-5 with common difference 60.

p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018

Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - M. F. Hasler, Oct 30 2018

Corrected by Jud McCranie, Jan 04 2001
a(11)-a(18) from Donovan Johnson, Sep 05 2008
Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018

A161534 The smallest of four consecutive primes where all three gaps are perfect squares.

Original entry on oeis.org

255763, 604441, 651361, 884497, 913063, 1065133, 1320211, 1526191, 2130133, 2376721, 2907727, 2911933, 2974891, 3190597, 3603583, 3690151, 3707497, 3962941, 4209643, 4245643, 4706101, 5057671, 5155567, 5223187, 5260711, 5321191, 5325571, 5410627

Offset: 1

Ki Punches, Jun 13 2009

nonn

Gaps occur as (36,4,36), (4,36,36), etc., all with at least one of them equal to 36 thru primes of 10^9.
A gap of 16 is first involved in 2376721 and 4706101, a gap of 64 first in 4245643, 5710531 and 21953641.
a(26) = 5321191 = A052239(6) = A058252(1) is the first term to be followed by three equal gaps, i.e., to start a sequence of consecutive primes in arithmetic progression (CPAP-4). - M. F. Hasler, Nov 06 2018

			a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A161002, A161533, A138198.

PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 3; p1 = 1; p2 = 2; p3 = 3; p4 = 5; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = p4; p4 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p4 - p3], AppendTo[t, p1]]]; t (* T. D. Noe, Jul 09 2013 *)
Transpose[Select[Partition[Prime[Range[400000]],4,1],And@@IntegerQ/@ Sqrt[ Differences[#]]&]][[1]] (* Harvey P. Dale, Mar 24 2014 *)

Terms beyond a(6) from R. J. Mathar, Sep 23 2009

A286817 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 90.

Original entry on oeis.org

4480114337, 11598050027, 16107995833, 19544227597, 19921527331, 26594502101, 29300624819, 29558893643, 30312236239, 30570499049, 31578774413, 38072416381, 38554165297, 43191763313, 44924731961, 45264825169, 45870718093, 46642295699, 47647807219, 48621561623

Offset: 1

Zak Seidov, May 27 2017

nonn

a(1) = 4480114337 = A052239(15).

Lars Blomberg, Table of n, a(n) for n = 1..10000

More terms from Lars Blomberg, Jun 06 2017

A287171 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 78.

Original entry on oeis.org

998051413, 4976990483, 5038640083, 5280622483, 5901887533, 6840045103, 8733200093, 11838052453, 11978292923, 12348742883, 12844988753, 13009792643, 13658681843, 13849573433

Offset: 1

Zak Seidov, May 27 2017

nonn

a(1) = 998051413 = A052239(13).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A052239.

Select[Partition[Prime[Range[7*10^8]],4,1],Differences[#]=={78,78,78}&][[All,1]] (* Harvey P. Dale, Mar 17 2018 *)

p=2;q=3;r=5; forprime(s=7,, if(s-r==78 && r-q==78 && q-p==78, print1(p", ")); p=q; q=r; r=s) \\ Charles R Greathouse IV, May 27 2017

a(n) >> n log^4 n. - Charles R Greathouse IV, May 28 2017

A287550 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.

Original entry on oeis.org

491525857, 1470227987, 2834347387, 4314407477, 4766711387, 6401372837, 6871241197, 8971400797, 10168905497, 11776429517, 11871902557, 14538547967, 14925896087, 15218517367, 15646776877, 15875854927, 17310026197, 17942416307, 18347931587, 19241492057, 19379888947

Offset: 1

Zak Seidov, May 26 2017

nonn

a(1)=491525857=A052239(12).

Chai Wah Wu, Table of n, a(n) for n = 1..100

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60]. Cf. A052239.

from gmpy2 import is_prime, next_prime
A287550_list, p = [], 2
q, r, s = p+72, p+144, p+216
while s <= 10**10:
    np = next_prime(p)
    if np == q and is_prime(r) and is_prime(s) and next_prime(q) == r and next_prime(r) == s:
        A287550_list.append(p)
    p, q, r, s = np, np+72, np+144, np+216 # Chai Wah Wu, Jun 03 2017

a(8)-a(21) from Chai Wah Wu, Jun 03 2017

A287593 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 84.

Original entry on oeis.org

2060959049, 5179429429, 5817066779, 12956411939, 13177258649, 13180695739, 13421782139, 14203265309, 14388905039, 14550841619, 14837398039, 16968792109, 20244234709

Offset: 1

Zak Seidov, May 27 2017

nonn

a(1) = 2060959049 = A052239(14).

A054203 a(n) is the smallest start of a run of exactly n+1 consecutive primes with n (not necessarily equal) prime differences, each divisible by 6.

Original entry on oeis.org

23, 47, 251, 1889, 1741, 19471, 118801, 498259, 148531, 406951, 1820111, 2339041, 40727657, 19725473, 73451737, 232301497, 400414121, 1057859471, 489144599, 1444257673, 766319189, 24061965043, 87996684091, 21549657539, 141116164769, 140432294381, 437339303279

Offset: 1

Labos Elemer, May 17 2000

nonn

This is a "modular arithmetic progression" of successive primes, modulo 6.

			For n = 1: a(1) = 23 is followed by a difference 6 to give 29, a prime.
For n = 5 a(5) = 1741 is followed by differences {6, 6, 6, 18, 6} and results in {1741, 1747, 1753, 1759, 1777, 1783} consecutive prime sequence.
For n = 10: a(10) = 406951 is prime prime is followed by {18, 12, 12, 30, 24, 12, 24, 36, 18, 12} consecutive differences pattern.

Index entries for sequences related to primes in arithmetic progressions.

Cf. A001223, A033451, A052243, A052239.

list(len) = {my(s = vector(len), v = [], prv = 2, c = 0, i, q, d); forprime(p = 3, , d = p - prv; if(d % 6, if(q > 0, i = #v; if(i > 0 && i <= len && s[i] == 0, s[i] = q; c++)); v = [], if(#v == 0, q = prv); v = concat(v, p)); prv = p; if(c == len, break)); s;} \\ Amiram Eldar, Mar 11 2025

a(11)-a(21) from Sean A. Irvine, Jan 25 2022

a(8) corrected, a(22)-a(27) added, and name clarified by Amiram Eldar, Mar 11 2025

cp's OEIS Frontend

A089236 Duplicate of A052239.

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A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

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A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

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A161534 The smallest of four consecutive primes where all three gaps are perfect squares.

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A286817 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 90.

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A287171 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 78.

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A287550 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.

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