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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)}.
1
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.
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 A375134 A295261
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