A054807 -id:A054807 - cp's OEIS frontend

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

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

A054806 Third 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

41, 71, 101, 227, 281, 311, 461, 487, 617, 641, 727, 757, 857, 881, 937, 1091, 1301, 1427, 1451, 1481, 1549, 1663, 1667, 1697, 1741, 1783, 1787, 1861, 1867, 1871, 1993, 1997, 2063, 2141, 2339, 2377, 2381, 2467, 2473, 2521, 2531, 2539, 2591, 2617, 2657

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..10000, 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 quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,3]] (* Harvey P. Dale, Aug 28 2021 *)

a(n) = nextprime(A054805(n)) = prevprime(A054807(n)), nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Offset corrected to 1 by M. F. Hasler, Oct 27 2018

Definition clarified by N. J. A. Sloane, Aug 28 2021

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

cp's OEIS Frontend

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

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A054806 Third term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

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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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