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%I #13 Sep 08 2022 08:45:46
%S 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,
%T 48,42,44,40,48,54,62,58,64,66,68,68,66,76,58,66,72,72,80,76,88,84,86,
%U 74,86,96,90,100,96,96,92,106,96,106,114,110,104,122,120,124,124,120,114
%N 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)}.
%C 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.
%F a(n) = 2*n - A058043(n). - _R. J. Mathar_, Jul 13 2009
%p 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
%t Table[2i - (Prime[PrimePi[i^2]+1]-Prime[PrimePi[i^2]]),{i,2,1000}]
%t 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 *)
%o (Magma) [2*n-(NthPrime(#PrimesUpTo(n^2)+1)-NthPrime(#PrimesUpTo(n^2))): n in [2..100]]; // _Vincenzo Librandi_, Aug 02 2015
%Y Cf. A058043.
%K nonn
%O 2,1
%A _Daniel Tisdale_, Jul 02 2009
%E Edited by _N. J. A. Sloane_, Jul 05 2009
%E Offset corrected by _R. J. Mathar_, Jul 13 2009
%E a(18) and further terms from _Robert G. Wilson v_, Aug 17 2009
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