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

1,2

Comments

Prime gaps are differences between adjacent prime numbers.

Also lengths of maximal arithmetic progressions of consecutive primes.

Links

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

Wikipedia, Arithmetic progression

Wikipedia, Longest increasing subsequence

Formula

Partial sums are A333214.

Example

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), ...

Maple

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

Mathematica

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

Crossrefs

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.

Sequence in context: A043286 A172098 A353801 * A204162 A266227 A043285

Adjacent sequences: A333251 A333252 A333253 * A333255 A333256 A333257

Keyword

nonn

Author

Gus Wiseman, Mar 20 2020

Status

approved