A054819 -id:A054819 - cp's OEIS frontend

A054809 Second term of strong prime 5-tuples: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

1657, 1777, 1847, 1861, 1987, 2371, 2459, 2503, 2521, 3433, 3449, 4201, 4507, 5261, 5407, 5431, 6029, 6637, 7229, 7283, 7741, 7867, 7919, 8147, 8501, 9587, 9601, 11027, 11369, 11579, 11821, 12391, 13859, 14813, 15121, 15527, 16033, 16301, 16811, 17011, 17377

Offset: 1

Henry Bottomley, Apr 10 2000

Initial member of pairs of consecutive primes in A054805 (second of quadruples): The first 10^4 terms of that sequence yield over 2000 terms of this sequence. - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..2000

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quadruples (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime 4-tuples, 5-tuples, 6-tuples; A054819 .. A054840: members of weak prime 4-tuples, ..., 7-tuples.

spqQ[n_]:=Module[{difs=Differences[n]},difs[[1]]>difs[[2]]> difs[[3]]> difs[[4]]]; Transpose[Select[Partition[Prime[ Range[2000]],5,1], spqQ]][[2]] (* Harvey P. Dale, May 06 2012 *)

a(n) = nextprime(A054808(n)) = prevprime(A054810(n)), nextprime = A151800, prevprime = A151799; A054809 = {m = A054805(n) | nextprime(m) = A054805(n+1)}. - M. F. Hasler, Oct 27 2018

Corrected by Harvey P. Dale, May 06 2012

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054810 Third term of strong prime 5-tuples: p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1).

Original entry on oeis.org

1663, 1783, 1861, 1867, 1993, 2377, 2467, 2521, 2531, 3449, 3457, 4211, 4513, 5273, 5413, 5437, 6037, 6653, 7237, 7297, 7753, 7873, 7927, 8161, 8513, 9601, 9613, 11047, 11383, 11587, 11827, 12401, 13873, 14821, 15131, 15541, 16057, 16319

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2000 (first 1001 terms by Harvey P. Dale).

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quadruples (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime 4-tuples, 5-tuples, 6-tuples; A054819 .. A054840: members of weak prime 4-tuples, ..., 7-tuples.

spqQ[{a_,b_,c_,d_,e_}]:=(b-a)>(c-b)>(d-c)>(e-d); Transpose[ Select[ Partition[ Prime[ Range[2000]],5,1],spqQ]][[3]] (* Harvey P. Dale, Feb 25 2013 *)

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054836 Third term of weak prime septet: p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3).

Original entry on oeis.org

15383, 64927, 68213, 68903, 128987, 128993, 143519, 154087, 158009, 192383, 221723, 222403, 244471, 249737, 285301, 318683, 337283, 354377, 357839, 374189, 385397, 394733, 402587, 402593, 419603, 439171, 441923, 448387, 457403, 457679, 458197, 482513, 527987, 529819, 577537, 582767

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..1000, Oct 27 2018.

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = A151800(A054835(n)) = A151799(A054838(n)), A151800 = nextprime, A151799 = prevprime; A054836 = { m = A054829(n) | m = nextprime(A054829(n-1)) }. - M. F. Hasler, Oct 27 2018

More terms from M. F. Hasler, Oct 27 2018

A054839 Sixth term of weak prime septet: p(m-4)-p(m-5) < p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

Original entry on oeis.org

15413, 64969, 68239, 68927, 129011, 129023, 143551, 154127, 158047, 192431, 221747, 222461, 244507, 249779, 285377, 318713, 337313, 354401, 357913, 374239, 385433, 394759, 402613, 402631, 419651, 439217, 441971, 448451, 457433, 457711, 458239, 482539, 528013

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..233, Oct 27 2018.

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

Transpose[Select[Partition[Prime[Range[50000]],7,1],Min[ Differences[ #,2]]> 0&]][[6]] (* Harvey P. Dale, Sep 27 2015 *)

a(n) = A151800(A054838(n)) = A151799(A054840(n)), A054839 = { m = A054832(n) | m = A151800(A054832(n-1)) } (A151800: nextprime, A151799: prevprime). - M. F. Hasler, Oct 27 2018

More terms from Harvey P. Dale, Sep 27 2015

A335277 First index of strictly increasing prime quartets.

Original entry on oeis.org

7, 13, 22, 28, 49, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 116, 122, 123, 144, 148, 152, 155, 173, 178, 182, 195, 201, 206, 212, 215, 219, 225, 226, 230, 236, 237, 244, 253, 256, 257, 265, 288, 302, 307, 315, 325, 328, 329, 332, 333, 336, 348, 355, 361, 373

Offset: 1

Gus Wiseman, May 30 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) < g(k + 1) < g(k + 2).

			The first 10 strictly increasing prime 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
For example, 107 is the 28th prime, and the primes (107,109,113,127) have differences (2,4,14), which are strictly increasing, so 28 is in the sequence.

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A335278.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime 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, A054800, A054819, A059044, A084758, A089180.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xPrimePi[x]]

prime(a(n)) = A054819(n).

A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

Original entry on oeis.org

1667, 1787, 1867, 1871, 1997, 2381, 2473, 2531, 2539, 3457, 3461, 4217, 4517, 5279, 5417, 5441, 6043, 6659, 7243, 7307, 7757, 7877, 7933, 8167, 8521, 9613, 9619, 11057, 11393, 11593, 11831, 12409, 13877, 14827, 15137, 15551, 16061, 16333

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

First member of pairs of consecutive primes in A054807 (4th of strong prime quartets). - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..2000, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

a(n) = nextprime(A054810(n)) = prevprime(A054812(n)), nextprime = A151800, prevprime = A151799; A054811 = {m = A054807(n) | nextprime(m) = A054807(n+1)}. - M. F. Hasler, Oct 27 2018

A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

Original entry on oeis.org

1669, 1789, 1871, 1873, 1999, 2383, 2477, 2539, 2543, 3461, 3463, 4219, 4519, 5281, 5419, 5443, 6047, 6661, 7247, 7309, 7759, 7879, 7937, 8171, 8527, 9619, 9623, 11059, 11399, 11597, 11833, 12413, 13879, 14831, 15139, 15559, 16063, 16339

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Second member of pairs of consecutive primes in A054807 (4th term of strong prime quartets). - M. F. Hasler, Oct 27 2018

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

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

spqQ[c_]:=Module[{d=Differences[c]},d[[1]]>d[[2]]>d[[3]]>d[[4]]]; Transpose[ Select[Partition[Prime[Range[2000]],5,1],spqQ]][[5]] (* Harvey P. Dale, Jan 01 2013 *)

a(n) = nextprime(A054811(n)); A054811 = {m = A054807(n) | prevprime(m) = A054807(n-1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

Original entry on oeis.org

1831, 2477, 3413, 9551, 21433, 22973, 25189, 26053, 32143, 33359, 33893, 39047, 40771, 41203, 44221, 47251, 48787, 55849, 57751, 66977, 70079, 74231, 74653, 74687, 75083, 75109, 82913, 84263, 87811, 88339, 88609, 103723, 103843, 106219, 106921, 108139, 110881, 112979, 118093

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = A151799(A054814(n)), A054813 = { m = A054808(n) | m = A151799(A054808(n+1)) }, where A151799 = next smaller prime. - M. F. Hasler, Oct 27 2018

More terms and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Original entry on oeis.org

1847, 2503, 3433, 9587, 21467, 22993, 25219, 26083, 32159, 33377, 33911, 39079, 40787, 41213, 44249, 47269, 48799, 55871, 57773, 67003, 70099, 74257, 74687, 74699, 75109, 75133, 82939, 84299, 87833, 88379, 88643, 103769, 103867, 106243, 106937, 108161, 110899, 112997, 118127, 120371

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Subsequence of A054808.

Select[Partition[Prime[Range[12000]],6,1],Max[Differences[#,2]]<0&][[;;,2]] (* Harvey P. Dale, Jun 17 2023 *)

a(n) = A151800(A054813(n)) = A151799(A054815(n)), A151800 = nextprime, A151799 = prevprime; A054814 = { m = A054809(n) | m = nextprime(A054809(n-1)) }. - M. F. Hasler, Oct 27 2018

Edited and offset changed to 1 by M. F. Hasler, Oct 26 2018

A054833 Sixth term of weak prime sextet: p(m-4)-p(m-5) < p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1).

Original entry on oeis.org

2953, 13513, 13931, 15013, 15413, 15427, 21467, 21587, 21911, 28307, 30931, 33287, 35671, 37781, 42281, 42641, 45853, 55691, 64709, 64969, 64997, 68239, 68261, 68927, 68947, 73999, 74257, 78643, 83177, 85571, 87383, 88069, 90971, 91621, 92297, 97073, 106853, 118529, 119083, 127807, 129011

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

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

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

Transpose[Select[Partition[Prime[Range[9000]],6,1],And@@Positive[ Differences[ #,2]]&]][[6]] (* Harvey P. Dale, Nov 06 2011 *)

a(n) = A151800(A054832(n)). - M. F. Hasler, Oct 27 2018

cp's OEIS Frontend

A054809 Second term of strong prime 5-tuples: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

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A054810 Third term of strong prime 5-tuples: p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1).

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A054836 Third term of weak prime septet: p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3).

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A054839 Sixth term of weak prime septet: p(m-4)-p(m-5) < p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m).

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A335277 First index of strictly increasing prime quartets.

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A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

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A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

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A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

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A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

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