A090832 -id:A090832 - cp's OEIS frontend

A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 20 2020

nonn

Prime gaps are differences between adjacent prime numbers.

Also lengths of maximal arithmetic progressions of consecutive primes.

			The prime gaps split into the following runs: (1), (2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), (2), (6), (4), ...

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic progression
Wikipedia, Longest increasing subsequence

The version for A000002 is A000002. Similarly for A001462.
The unequal version is A333216.
The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
Positions of first appearances are A335406.
The first term of the first length-n arithmetic progression of consecutive primes is A006560(n), with index A089180(n).
Prime gaps are A001223.
Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Cf. A000040, A031217, A054800, A059044, A084758, A090832, A124767, A238279, A295235.

p:= 3: t:= 1: R:= NULL: s:= 1: count:= 0:
for i from 2 while count < 100 do
  q:= nextprime(p);
  g:= q-p; p:= q;
  if g = t then s:= s+1
  else count:= count+1; R:= R, s; t:= g; s:= 1;
  fi
od:
R; # Robert Israel, Jan 06 2021

Length/@Split[Differences[Array[Prime,100]],#1==#2&]//Most

Partial sums are A333214.

A033451 Initial prime in set of 4 consecutive primes with common difference 6.

Original entry on oeis.org

251, 1741, 3301, 5101, 5381, 6311, 6361, 12641, 13451, 14741, 15791, 15901, 17471, 18211, 19471, 23321, 26171, 30091, 30631, 53611, 56081, 62201, 63691, 71341, 75521, 77551, 78791, 80911, 82781, 83431, 84431, 89101, 89381, 91291, 94421

Offset: 1

Jeff Burch

nonn

Primes p such that p, p+6, p+12, p+18 are consecutive primes.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of March 2013 the record is 10 primes.
Note that the Green and Tao reference is about arithmetic progressions that are not necessarily consecutive. - Michael B. Porter, Mar 05 2013
Subsequence of A023271. - R. J. Mathar, Nov 04 2006
All terms p == 1 (mod 10) and hence p+24 are always divisible by 5. - Zak Seidov, Jun 20 2015
Subsequence of A054800, with which is coincides up to a(24), but a(25) = A054800(26). - M. F. Hasler, Oct 26 2018

			251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.

T. D. Noe, Table of n, a(n) for n = 1..1000
Jens Kruse Andersen, The Largest Known CPAP's
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007.
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Intersection of A054800 and A023271.
Cf. A090832, A090833, A090834, A090835, A090836, A090837, A090838, A090839.
Analogous sequences [with common difference in square brackets]: A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388[48].
Cf. A058362, A059044.
Subsequence of A047948.

N:=10^5: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+18,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
Primes[t+3]-Primes[t+2]]=[6,6,6], [$1..nops(Primes)-3])]; # Muniru A Asiru, Aug 04 2017

A033451 = Reap[ For[p = 2, p < 100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 6, p3 = NextPrime[p2]; If[p3 - p2 == 6, p4 = NextPrime[p3]; If[p4 - p3 == 6, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[16000]],4,1],Union[ Differences[ #]] == {6}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

p=2;q=3;r=5;forprime(s=7,1e4,if(s-p==18 && s-q==12 && s-r==6, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A000040(A090832(n)). - Zak Seidov, Jun 20 2015

A054819 First term of weak prime quartet: p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2).

Original entry on oeis.org

17, 41, 79, 107, 227, 281, 311, 347, 349, 379, 397, 439, 461, 499, 569, 641, 673, 677, 827, 857, 881, 907, 1031, 1061, 1091, 1187, 1229, 1277, 1301, 1319, 1367, 1427, 1429, 1451, 1487, 1489, 1549, 1607, 1619, 1621, 1697, 1877, 1997, 2027, 2087, 2153

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

			From _Gus Wiseman_, May 31 2020: (Start)
The first 10 strictly increasing prime gap quartets:
   17   19   23   29
   41   43   47   53
   79   83   89   97
  107  109  113  127
  227  229  233  239
  281  283  293  307
  311  313  317  331
  347  349  353  359
  349  353  359  367
  379  383  389  397
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051635, A054800-A054840.
Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime gap quartets are A335278.
Strictly increasing prime gap quartets are A335277.
Equal prime gap quartets are A090832.
Weakly increasing prime gap quartets are A333383.
Weakly decreasing prime gap quartets are A333488.
Unequal prime gap quartets are A333490.
Partially unequal prime gap quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Lengths of maximal weakly decreasing sequences of prime gaps are A333212.
Lengths of maximal strictly increasing sequences of prime gaps are A333253.
Cf. A000040, A006560, A031217, A059044, A084758, A089180, A333253.

wpqQ[lst_]:=Module[{diffs=Differences[lst]},diffs[[1]]Harvey P. Dale, Jun 12 2012 *)
ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xx] (* Gus Wiseman, May 31 2020 *)

a(n) = prime(A335277(n)). - Gus Wiseman, May 31 2020

A054804 First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

31, 61, 89, 211, 271, 293, 449, 467, 607, 619, 709, 743, 839, 863, 919, 1069, 1291, 1409, 1439, 1459, 1531, 1637, 1657, 1669, 1723, 1759, 1777, 1831, 1847, 1861, 1979, 1987, 2039, 2131, 2311, 2357, 2371, 2447, 2459, 2477, 2503, 2521, 2557, 2593, 2633

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Primes preceding the first member of pairs of consecutive primes in A051634 ("strong primes"), see example. (A051634 lists the middle member of the triplets, here we list the first member of the quadruplets.) - M. F. Hasler, Oct 27 2018, corrected thanks to Gus Wiseman, Jun 01 2020.

			The first 10 strictly decreasing prime gap quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence.
The second and third term of each quadruplet are consecutive terms in A051634: this is a characteristic property of this sequence. - _M. F. Hasler_, Jun 01 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051634, A054800-A054840.
Prime gaps are A001223.
Second prime gaps are A036263.
All of the following use prime indices rather than the primes themselves:
- Strictly decreasing prime gap quartets are A335278.
- Strictly increasing prime gap quartets are A335277.
- Equal prime gap quartets are A090832.
- Weakly increasing prime gap quartets are A333383.
- Weakly decreasing prime gap quartets are A333488.
- Unequal prime gap quartets are A333490.
- Partially unequal prime gap quartets are A333491.
- Adjacent equal prime gaps are A064113.
- Strict ascents in prime gaps are A258025.
- Strict descents in prime gaps are A258026.
- Adjacent unequal prime gaps are A333214.
- Weak ascents in prime gaps are A333230.
- Weak descents in prime gaps are A333231.
Maximal weakly increasing intervals of prime gaps are A333215.
Maximal strictly decreasing intervals of prime gaps are A333252.
Cf. also A000040, A006560, A031217, A054800, A054805, A054806, A054807, A059044, A084758, A089180, A333253, A335278.

primes:= select(isprime,[seq(i,i=3..10000,2)]):
L:=  primes[2..-1]-primes[1..-2]:
primes[select(t -> L[t+2] < L[t+1] and L[t+1] < L[t], [$1..nops(L)-2])]; # Robert Israel, Jun 28 2018

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>x] (* Gus Wiseman, May 31 2020 *)
Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* Harvey P. Dale, Jan 12 2023 *)

a(n) = prime(A335278(n)). - Gus Wiseman, May 31 2020

A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

Original entry on oeis.org

2, 13, 21, 3, 7, 8, 1, 18, 29, 5, 3, 8, 11, 31, 4, 20, 3, 7, 5, 19, 21, 32, 1, 19, 48, 19, 29, 32, 7, 38, 1, 43, 12, 33, 46, 6, 16, 8, 4, 34, 15, 1, 19, 7, 1, 23, 28, 30, 22, 8, 1, 7, 1, 52, 14, 56, 10, 26, 2, 30, 65, 5, 71, 12, 44, 39, 37, 6, 19, 47, 11, 10

Offset: 1

Gus Wiseman, Mar 15 2020

Prime gaps are differences between adjacent prime numbers.

Essentially the same as A145024. - R. J. Mathar, Mar 16 2020

			The prime gaps split into the following subsequences without adjacent equal terms: (1,2), (2,4,2,4,2,4,6,2,6,4,2,4,6), (6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6), (6,4,6), (6,2,10,2,4,2,12), (12,4,2,4,6,2,10,6), ...

First differences of A064113.
The version for the Kolakoski sequence is A306323.
The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A001223, A084758, A106356, A124762, A124767, A333214.

Length/@Split[Differences[Array[Prime,100]],UnsameQ]//Most

Ones correspond to balanced prime quartets (A054800), so the sum of terms up to but not including the n-th one is A000720(A054800(n - 1)) = A090832(n).

A090839 Numbers k such that 6k+1, 6k+7, 6k+13, 6k+19 are consecutive primes.

Original entry on oeis.org

290, 550, 850, 1060, 2650, 3035, 3245, 5015, 5105, 8935, 10615, 11890, 12925, 13485, 13905, 14850, 15215, 15985, 17560, 17600, 18105, 19925, 20135, 21780, 23510, 24040, 25490, 28830, 31145, 34365, 36355, 38140, 38370, 42025, 43845, 46820, 47575, 48745, 49130, 50495, 53350

Offset: 1

Pierre CAMI, Dec 09 2003

All terms are == 0 (mod 5). - Robert G. Wilson v, Dec 12 2017

			6*290 + 1 = 1741, 6*290 + 7 = 1747, 6*290 + 13 = 1753, 6*290 + 19 = 1759 and 1741, 1747, 1753, 1759 are consecutive primes, so 290 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033451, A090832, A090833, A090834, A090835, A090836, A090837, A090838.

Block[{nn = 50500, s}, s = Select[Prime@ Range@ PrimePi[6 (nn + 3) - 1], Divisible[(# + 1), 6] &]; Select[Range@ nn, And[AllTrue[#, PrimeQ], Count[s, q_ /; First[#] < q < Last@ #] == 0] &@ Map[6 # + 1 &, # + Range[0, 3]] &]] (* Michael De Vlieger, Dec 06 2017 *)
fQ[n_] := Block[{p = {6n +1, 6n +7, 6n +13, 6n +19}}, Union@ PrimeQ@ p == {True} && NextPrime[6n +1, 3] == 6n +19]; Select[5 Range@ 10100, fQ] (* Robert G. Wilson v, Dec 12 2017 *)
Select[(#-1)/6&/@Select[Partition[Prime[Range[30000]],4,1],Differences[#]=={6,6,6}&][[;;,1]],IntegerQ] (* Harvey P. Dale, Apr 05 2025 *)

isok(n) = my(p,q,r); isprime(p=6*n+1) && ((q=6*n+7) == nextprime(p+1)) && ((r=6*n+13) == nextprime(q+1)) && (6*n+19 == nextprime(r+1)); \\ Michel Marcus, Sep 20 2019

Missing term 5105 and more terms from Michel Marcus, Sep 20 2019

A090836 Numbers n such that 6n+5, 6n+11, 6n+17, 6n+23 are consecutive primes.

Original entry on oeis.org

41, 896, 1051, 2106, 2241, 2456, 2631, 2911, 3886, 4361, 9346, 10366, 12586, 13131, 13796, 14071, 14896, 15736, 15876, 17451, 19291, 20091, 20166

Offset: 1

Pierre CAMI, Dec 09 2003

			6*41+5=251, 6*41+11=257, 6*41+17=263, 6*41+23=269; 251,257,263,269 are consecutive primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A033451, A090832, A090833, A090834, A090835, A090837, A090838, A090839.

Select[(#-5)/6&/@Transpose[Select[Partition[Prime[Range[11500]],4,1], Union[Differences[#]]=={6}&]][[1]],IntegerQ] (* Harvey P. Dale, Nov 18 2013 *)

A090833 Numbers n such that 6n+5, 6n+11, 6n+17, 6n+23 are consecutive primes or 6n+1, 6n+7, 6n+13, 6n+19 are consecutive primes.

Original entry on oeis.org

41, 290, 550, 850, 896, 1051, 1060, 2106, 2241, 2456, 2631, 2650, 2911, 3035, 3245, 3886, 4361, 5015, 5105, 8935, 9346, 10366, 10615, 11890, 12586, 12925, 13131, 13485, 13796, 13905, 14071, 14850, 14896, 15215, 15736, 15876, 15985, 17451, 17560

Offset: 1

Pierre CAMI, Dec 09 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A033451, A090832, A090834, A090835, A090836, A090837, A090838, A090839.

Select[Range[20000],(PrimeQ[6#+5]&&Differences[NestList[ NextPrime[ #]&, 6 #+5,3]] =={6,6,6})||(PrimeQ[6#+1]&&Differences[NestList[NextPrime[#]&,6 #+1,3]]=={6,6,6})&] (* Harvey P. Dale, Sep 23 2016 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==18&&s-q==12&&s-r==6,print1(p\6", "));p=q;q=r;r=s) \\ Charles R Greathouse IV, Dec 27 2011

a(19) from Charles R Greathouse IV, Dec 27 2011

A090834 Primes p such that p, p+6, p+12, p+18 are consecutive primes and p=6*k+5 for some k.

Original entry on oeis.org

251, 5381, 6311, 12641, 13451, 14741, 15791, 17471, 23321, 26171, 56081, 62201, 75521, 78791, 82781, 84431, 89381, 94421, 95261, 104711, 115751, 120551, 121001, 154061, 162251, 163841, 179801, 185051, 187361, 191021, 206021, 214451, 222311, 226631, 243521

Offset: 1

Pierre CAMI, Dec 09 2003

			251,257,263,269 are consecutive primes,257=251+6,263=251+12,269=251+18 and 251=6*41+5

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A033451, A090832, A090833, A090835, A090836, A090837, A090838, A090839.

Transpose[Select[Partition[Prime[Range[30000]],4,1],Differences[#]=={6,6,6}&&IntegerQ[(#[[1]]-5)/6]&]][[1]] (* Harvey P. Dale, Dec 12 2015 *)

More terms from Harvey P. Dale, Dec 12 2015

A090837 Primes p such that p, p+6, p+12, p+18 are consecutive primes and p = 6*k+1 for some k.

Original entry on oeis.org

1741, 3301, 5101, 6361, 15901, 18211, 19471, 30091, 30631, 53611, 63691, 71341, 77551, 80911, 83431, 89101, 91291, 95911, 105361, 105601, 108631, 119551, 120811, 130681, 141061, 144241, 152941, 172981, 186871, 206191, 218131, 228841, 230221, 252151, 263071, 280921, 285451

Offset: 1

Pierre CAMI, Dec 09 2003

			1741, 1747, 1753, 1759 are consecutive primes, 1747 = 1741 + 6, 1753 = 1741 + 12, 1759 = 1741 + 18 and 1741 = 6 * 290 + 1.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A033451, A090832, A090833, A090834, A090835, A090836, A090838, A090839.

filter:= p -> isprime(p) and nextprime(p) = p+6 and nextprime(p+6)=p+12 and nextprime(p+12)=p+18:
select(filter, [seq(i,i=1..10^6,6)]); # Robert Israel, Nov 11 2020

isok(p) = my(q,r,s); isprime(p) && ((p % 6) == 1) && ((q=nextprime(p+1)) == p+6) && ((r=nextprime(q+1)) == p+12) && ((s=nextprime(r+1)) == p+18); \\ Michel Marcus, Sep 20 2019

More terms from Michel Marcus, Sep 20 2019

cp's OEIS Frontend

A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A033451 Initial prime in set of 4 consecutive primes with common difference 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A054819 First term of weak prime quartet: p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A054804 First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A090839 Numbers k such that 6*k+1, 6*k+7, 6*k+13, 6*k+19 are consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A090836 Numbers n such that 6*n+5, 6*n+11, 6*n+17, 6*n+23 are consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A090833 Numbers n such that 6n+5, 6n+11, 6n+17, 6n+23 are consecutive primes or 6n+1, 6n+7, 6n+13, 6n+19 are consecutive primes.

A090839 Numbers k such that 6k+1, 6k+7, 6k+13, 6k+19 are consecutive primes.

A090836 Numbers n such that 6n+5, 6n+11, 6n+17, 6n+23 are consecutive primes.