A038529 n-th prime - n-th composite.

-2, -3, -3, -2, 1, 1, 3, 4, 7, 11, 11, 16, 19, 19, 22, 27, 32, 33, 37, 39, 40, 45, 48, 53, 59, 62, 63, 65, 65, 68, 81, 83, 88, 89, 98, 99, 103, 108, 111, 116, 121, 121, 129, 130, 133, 134, 145, 155, 158, 159, 161, 165, 166, 175, 180, 185, 189, 190, 195, 197, 198, 207

Offset: 1

Author

Vasiliy Danilov (danilovv(AT)usa.net), Jul 14 1998

Keywords

• sign
• easy

Comments

Sequence is monotonically increasing starting from a(2). a(n) = a(n+1) if and only if both prime(n)+2 and composite(n)+1 are prime. - Jianing Song, Jun 27 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014237, A067563.

Programs

Haskell

a038529 n = a000040 n - a002808 n  -- Reinhard Zumkeller, Apr 30 2014

Mathematica

composite[n_Integer] := Block[{k=n+PrimePi[n]+1}, While[k-PrimePi[k]-1 != n, k++]; k]; Table[Prime[n] - composite[n], {n,65}] (* corrected by Harvey P. Dale, Aug 08 2011 *)
Module[{nn=300,prs,cmps,len},prs=Prime[Range[PrimePi[nn]]];cmps= Complement[ Range[4,nn],prs];len=Min[Length[prs],Length[cmps]]; #[[1]]- #[[2]]&/@ Thread[{Take[prs,len],Take[cmps,len]}]] (* Harvey P. Dale, Jun 18 2015 *)

Python

from sympy import prime, composite
def A038529(n):
    return prime(n)-composite(n) # Chai Wah Wu, Dec 27 2018

Formula

a(n) = A000040(n) - A002808(n). - Reinhard Zumkeller, Apr 30 2014