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Differences between consecutive primes (A001223) but with repeats omitted.
40

%I #35 Sep 28 2024 19:04:26

%S 1,2,4,2,4,2,4,6,2,6,4,2,4,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,

%T 6,4,6,2,10,2,4,2,12,4,2,4,6,2,10,6,2,6,4,2,10,14,4,2,4,14,6,10,2,4,6,

%U 8,6,4,6,8,4,8,10,2,10,2,6,4,6,8,4

%N Differences between consecutive primes (A001223) but with repeats omitted.

%C Also the run-compression of the sequence of first differences of prime numbers, where 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, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1). - _Gus Wiseman_, Sep 16 2024

%H Reinhard Zumkeller, <a href="/A037201/b037201.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%F a(n>1) = 2*A373947(n-1). - _Gus Wiseman_, Sep 16 2024

%t Flatten[Split[Differences[Prime[Range[150]]]]/.{(k_)..}:>k] (* based on a program by Harvey P. Dale, Jun 21 2012 *)

%o (Haskell)

%o a037201 n = a037201_list !! (n-1)

%o a037201_list = f a001223_list where

%o f (x:xs@(x':_)) | x == x' = f xs

%o | otherwise = x : f xs

%o -- _Reinhard Zumkeller_, Feb 27 2012

%o (PARI) t=0;p=2;forprime(q=3,1e3,if(q-p!=t,print1(q-p", "));t=q-p;p=q) \\ _Charles R Greathouse IV_, Feb 27 2012

%Y This is the run-compression of A001223 = first differences of A000040.

%Y The repeats were at positions A064113 before being omitted.

%Y Adding up runs instead of compressing them gives A373822.

%Y The even terms halved are A373947.

%Y For prime-powers instead of prime numbers we have A376308.

%Y Positions of first appearances are A376520, sorted A376521.

%Y A003242 counts compressed compositions.

%Y A333254 lists run-lengths of differences between consecutive primes.

%Y A373948 encodes compression using compositions in standard order.

%Y Cf. A007921, A030173, A053289, A106356, A114901, A116608, A116861, A124767, A238130, A333755, A335406, A373954.

%K easy,nonn,nice

%O 1,2

%A _Naohiro Nomoto_

%E Offset corrected by _Reinhard Zumkeller_, Feb 27 2012