A162417 Find max {primes such that p < n^2, n = 2,3,...}, then the gap g(n) between that prime and its successor. This sequence is the sequence of differences {2n - g(n)}.

2, 2, 4, 4, 6, 8, 10, 14, 16, 8, 14, 20, 24, 26, 26, 24, 22, 30, 36, 38, 36, 28, 42, 38, 48, 48, 42, 44, 40, 48, 54, 62, 58, 64, 66, 68, 68, 66, 76, 58, 66, 72, 72, 80, 76, 88, 84, 86, 74, 86, 96, 90, 100, 96, 96, 92, 106, 96, 106, 114, 110, 104, 122, 120, 124, 124, 120, 114

Offset: 2

Author

Daniel Tisdale, Jul 02 2009

Keywords

• nonn

Comments

The unproved conjecture that 2n - g(n) > 0 would imply Legendre's conjecture, since the next prime after max {p < n^2} will always occur before (n+1)^2.

Crossrefs

Cf. A058043.

Programs

Magma

[2*n-(NthPrime(#PrimesUpTo(n^2)+1)-NthPrime(#PrimesUpTo(n^2))): n in [2..100]]; // Vincenzo Librandi, Aug 02 2015

Maple

with(numtheory): A162417:=n->2*n-(ithprime(pi(n^2)+1)-ithprime(pi(n^2))): seq(A162417(n), n=2..100); # Wesley Ivan Hurt, Aug 01 2015

Mathematica

Table[2i - (Prime[PrimePi[i^2]+1]-Prime[PrimePi[i^2]]),{i,2,1000}]
f[n_] := 2 n - Prime[PrimePi[n^2] + 1] + Prime[PrimePi[n^2]]; Table[ f@n, {n, 2, 69}] (* Robert G. Wilson v, Aug 17 2009 *)

Formula

a(n) = 2*n - A058043(n). - R. J. Mathar, Jul 13 2009

Extensions

Edited by N. J. A. Sloane, Jul 05 2009

Offset corrected by R. J. Mathar, Jul 13 2009

a(18) and further terms from Robert G. Wilson v, Aug 17 2009