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A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).

%I A333254 #14 Jan 06 2021 19:19:24
%S A333254 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,
%T A333254 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,
%U A333254 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).
%C A333254 Prime gaps are differences between adjacent prime numbers.
%C A333254 Also lengths of maximal arithmetic progressions of consecutive primes.
%H A333254 Robert Israel, <a href="/A333254/b333254.txt">Table of n, a(n) for n = 1..10000</a>
%H A333254 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%H A333254 Wikipedia, <a href="https://en.wikipedia.org/wiki/Longest_increasing_subsequence">Longest increasing subsequence</a>
%F A333254 Partial sums are A333214.
%e A333254 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), ...
%p A333254 p:= 3: t:= 1: R:= NULL: s:= 1: count:= 0:
%p A333254 for i from 2 while count < 100 do
%p A333254   q:= nextprime(p);
%p A333254   g:= q-p; p:= q;
%p A333254   if g = t then s:= s+1
%p A333254   else count:= count+1; R:= R, s; t:= g; s:= 1;
%p A333254   fi
%p A333254 od:
%p A333254 R; # _Robert Israel_, Jan 06 2021
%t A333254 Length/@Split[Differences[Array[Prime,100]],#1==#2&]//Most
%Y A333254 The version for A000002 is A000002. Similarly for A001462.
%Y A333254 The unequal version is A333216.
%Y A333254 The weakly decreasing version is A333212.
%Y A333254 The weakly increasing version is A333215.
%Y A333254 The strictly decreasing version is A333252.
%Y A333254 The strictly increasing version is A333253.
%Y A333254 Positions of first appearances are A335406.
%Y A333254 The first term of the first length-n arithmetic progression of consecutive primes is A006560(n), with index A089180(n).
%Y A333254 Prime gaps are A001223.
%Y A333254 Positions of adjacent equal prime gaps are A064113.
%Y A333254 Positions of adjacent unequal prime gaps are A333214.
%Y A333254 Cf. A000040, A031217, A054800, A059044, A084758, A090832, A124767, A238279, A295235.
%K A333254 nonn
%O A333254 1,2
%A A333254 _Gus Wiseman_, Mar 20 2020