A373947 - OEIS

%I #8 Jun 30 2024 09:10:10

%S 1,2,1,2,1,2,3,1,3,2,1,2,3,1,3,2,1,3,2,3,4,2,1,2,1,2,7,2,3,1,5,1,3,2,

%T 3,1,5,1,2,1,6,2,1,2,3,1,5,3,1,3,2,1,5,7,2,1,2,7,3,5,1,2,3,4,3,2,3,4,

%U 2,4,5,1,5,1,3,2,3,4,2,1,2,6,4,2,4,2,3

%N Halved and run-compressed first differences of consecutive odd primes.

%C We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

%F a(n) = A037201(n+1)/2.

%e The odd primes begin:

%e 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...

%e with differences:

%e 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, ...

%e with run-compression:

%e 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...

%e which is 2*a(n).

%t First/@Split[Differences[Select[Range[3,100],PrimeQ]]]/2

%Y Half of A037201(n+1).

%Y Run-compression of A001223.

%Y A003242, A114901, A116608, A116861, A238279, A240085, A333755, A373948, A373949, A373951, A373953, A373954.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 29 2024