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

2,1

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.

Links

Table of n, a(n) for n=2..69.

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A058043.

Sequence in context: A294150 A087135 A227135 * A240012 A295261 A293627

Adjacent sequences: A162414 A162415 A162416 * A162418 A162419 A162420

Keyword

nonn

Author

Daniel Tisdale, Jul 02 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

Status

approved