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A117490 Number of primes between n and n^2 (with n and n^2 excluded).

%I A117490 #21 Sep 07 2024 08:53:42
%S A117490 0,1,2,4,6,8,11,14,18,21,25,29,33,38,42,48,54,59,64,70,77,84,90,96,
%T A117490 105,113,120,128,136,144,151,161,170,180,189,199,207,216,228,239,250,
%U A117490 261,269,281,292,305,314,327,342,352,363,378,393,405,418,429,441,458,470
%N A117490 Number of primes between n and n^2 (with n and n^2 excluded).
%C A117490 A famous Japanese mathematics book states that this sequence is nonzero (for n>1) if the Riemann Hypothesis is true, but this statement seems to be false.
%C A117490 If the n-th prime is denoted by p(n) then a(j) = number of nonzero values of floor (j^2/p(n)), over all n >= 1, (derived from A165974). - _Christopher Hunt Gribble_, Oct 03 2009
%H A117490 Amiram Eldar, <a href="/A117490/b117490.txt">Table of n, a(n) for n = 1..10000</a>
%F A117490 a(n) = pi(n^2) - pi(n), cf. A000720.
%F A117490 a(n) = A038107(n) - A000720(n) = A073882(n) - A010051(n). - _Reinhard Zumkeller_, May 20 2010
%e A117490 For n = 5: between 5+1 = 6 and 5^2-1 = 24 there are the following six primes: 7, 11, 13, 17, 19, 23.
%p A117490 P:=proc(n) local i,j,np; for i from 1 by 1 to n do np:=0; for j from i+1 by 1 to i^2-1 do if isprime(j) then np:=np+1; fi; od; print(np); od; end: P(100);
%t A117490 a[n_] := PrimePi[n^2 - 1] - PrimePi[n]; Array[a, 59] (* _Robert G. Wilson v_, Apr 06 2006 *)
%Y A117490 Cf. A000720, A010051, A014085, A038107, A060715, A073882, A117491, A165974.
%K A117490 easy,nonn
%O A117490 1,3
%A A117490 _Paolo P. Lava_ and _Giorgio Balzarotti_, Mar 22 2006