A158038 Difference between n-th prime and next cube.

6, 5, 3, 1, 16, 14, 10, 8, 4, 35, 33, 27, 23, 21, 17, 11, 5, 3, 58, 54, 52, 46, 42, 36, 28, 24, 22, 18, 16, 12, 89, 85, 79, 77, 67, 65, 59, 53, 49, 43, 37, 35, 25, 23, 19, 17, 5, 120, 116, 114, 110, 104, 102, 92, 86, 80, 74, 72, 66, 62, 60, 50, 36, 32, 30, 26, 12, 6, 165, 163

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

Keywords

• nonn

Comments

Could be read as a table, since there are always several primes between two cubes. Whenever a(n+1) > a(n), the n-th prime is the largest one below a given cube and prime(n+1) is the smallest prime larger than that cube. For n > 1, these are also the indices where the parity of the terms changes. - M. F. Hasler, Oct 19 2018

Examples

			The first terms are: 8 - 2 = 6, 8 - 3 = 5, 8 - 5 = 3, 8 - 7 = 1, 27 - 11 = 16, ...
From _M. F. Hasler_, Oct 19 2018: (Start)
Starting a new row when going to the next cube, the sequence reads:
  6, 5, 3, 1,                      // = 8 - {primes between 1^3 = 1 and 2^3 = 8}
  16, 14, 10, 8, 4,                // = 27 - {primes between 2^3 = 8 and 3^3 = 27}
  35, 33, 27, 23, 21, 17, 11, 5, 3, // = 64 - {primes between 27 and 4^3 = 64}
  58, 54, 52, ..., 18, 16, 12,     // = 125 - {primes between 64 and 5^3 = 125}
  89, 85, 79, ..., 19, 17, 5,      // = 216 - {primes between 125 and 6^3 = 216}
  120, 116, 114, ..., 26, 12, 6,   // = 343 - {primes between 216 and 7^3 = 343}
  etc. (End)
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A106044, A104492, A056892, A158037

Programs

Mathematica

lst={};Do[p=Prime[n];s=p^(1/3);f=Floor[s];a=(f+1)^3;d=a-p;AppendTo[lst,d],{n,6!}];lst
nc[n_]:=(Floor[Surd[n,3]]+1)^3-n; Table[nc[n],{n,Prime[Range[70]]}] (* Harvey P. Dale, Jun 19 2014 *)

PARI

A158038(n)=(sqrtnint(n=prime(n),3)+1)^3-n \\ M. F. Hasler, Oct 19 2018

PARI

first(n) = my(res = vector(n), t = 0, c = 2, c3 = 8); forprime(p = 2, oo, t++; if(p > c3, c++; c3 = c^3); res[t] = c3 - p; if(t==n, return(res))) \\ David A. Corneth, Oct 19 2018

Formula

a(n) > 0. - David A. Corneth, Oct 19 2018

Extensions

Edited by M. F. Hasler, Oct 19 2018