A054819 - OEIS

A054819

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

%I #13 Jun 19 2020 03:58:36

%S 17,41,79,107,227,281,311,347,349,379,397,439,461,499,569,641,673,677,

%T 827,857,881,907,1031,1061,1091,1187,1229,1277,1301,1319,1367,1427,

%U 1429,1451,1487,1489,1549,1607,1619,1621,1697,1877,1997,2027,2087,2153

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

%H Harvey P. Dale, <a href="/A054819/b054819.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = prime(A335277(n)). - _Gus Wiseman_, May 31 2020

%e From _Gus Wiseman_, May 31 2020: (Start)

%e The first 10 strictly increasing prime gap quartets:

%e 17 19 23 29

%e 41 43 47 53

%e 79 83 89 97

%e 107 109 113 127

%e 227 229 233 239

%e 281 283 293 307

%e 311 313 317 331

%e 347 349 353 359

%e 349 353 359 367

%e 379 383 389 397

%e (End)

%t wpqQ[lst_]:=Module[{diffs=Differences[lst]},diffs[[1]]<diffs[[2]]<diffs[[3]]]; Transpose[Select[Partition[Prime[ Range[400]], 4,1],wpqQ]][[1]] (* _Harvey P. Dale_, Jun 12 2012 *)

%t ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x<z-y<t-z:>x] (* _Gus Wiseman_, May 31 2020 *)

%Y Cf. A051635, A054800-A054840.

%Y Prime gaps are A001223.

%Y Second prime gaps are A036263.

%Y Strictly decreasing prime gap quartets are A335278.

%Y Strictly increasing prime gap quartets are A335277.

%Y Equal prime gap quartets are A090832.

%Y Weakly increasing prime gap quartets are A333383.

%Y Weakly decreasing prime gap quartets are A333488.

%Y Unequal prime gap quartets are A333490.

%Y Partially unequal prime gap quartets are A333491.

%Y Positions of adjacent equal prime gaps are A064113.

%Y Positions of strict ascents in prime gaps are A258025.

%Y Positions of strict descents in prime gaps are A258026.

%Y Positions of adjacent unequal prime gaps are A333214.

%Y Positions of weak ascents in prime gaps are A333230.

%Y Positions of weak descents in prime gaps are A333231.

%Y Lengths of maximal weakly decreasing sequences of prime gaps are A333212.

%Y Lengths of maximal strictly increasing sequences of prime gaps are A333253.

%Y Cf. A000040, A006560, A031217, A059044, A084758, A089180, A333253.

%K nonn

%O 1,1

%A _Henry Bottomley_, Apr 10 2000