A037201 Differences between consecutive primes (A001223) but with repeats omitted.

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, 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, 8, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4

Offset: 1

Author

Naohiro Nomoto

Keywords

• easy
• nonn
• nice
• changed

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for primes, gaps between

Crossrefs

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

The repeats were at positions A064113 before being omitted.

Adding up runs instead of compressing them gives A373822.

The even terms halved are A373947.

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

Positions of first appearances are A376520, sorted A376521.

A003242 counts compressed compositions.

A333254 lists run-lengths of differences between consecutive primes.

A373948 encodes compression using compositions in standard order.

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

Programs

Haskell

a037201 n = a037201_list !! (n-1)
a037201_list = f a001223_list where
   f (x:xs@(x':_)) | x == x'   = f xs
                   | otherwise = x : f xs
-- Reinhard Zumkeller, Feb 27 2012

Mathematica

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

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

Formula

a(n>1) = 2*A373947(n-1). - Gus Wiseman, Sep 16 2024

Extensions

Offset corrected by Reinhard Zumkeller, Feb 27 2012