OFFSET
1,1
COMMENTS
Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) >= g(k + 1) >= g(k + 2).
EXAMPLE
The first 10 weakly decreasing prime quartets:
31 37 41 43
47 53 59 61
61 67 71 73
89 97 101 103
151 157 163 167
167 173 179 181
199 211 223 227
211 223 227 229
241 251 257 263
251 257 263 269
For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 is in the sequence.
MATHEMATICA
ReplaceList[Array[Prime, 100], {___, x_, y_, z_, t_, ___}/; y-x>=z-y>=t-z:>PrimePi[x]]
CROSSREFS
Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488 (this sequence).
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 subsequences of prime gaps: A333212.
Lengths of maximal strictly increasing subsequences of prime gaps: A333253.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2020
STATUS
approved