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%I #13 Mar 04 2019 18:47:29
%S 0,0,0,1,0,1,0,1,2,1,0,1,2,2,1,0,0,1,2,2,2,2,2,2,2,1,0,0,1,2,3,3,3,3,
%T 3,2,2,2,1,0,1,1,1,2,2,3,2,2,2,2,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,1,2,2,
%U 2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,5,4,4,4,4,3,3,2,1,0,0,1,1,1,1,2,2,2,2
%N Number of semiprimes between n and n + sqrt(n).
%C It appears that for n > 37 it is always true that a(n) > 0. The exponent can be reduced further. Since 597 + 597^(0.4129) > 611, leaping the record semiprime gap between 597 and 611, it seems that for n > 597 it is always true that there is a semiprime between n and n^(0.4129). It seems that for n > 2705 it is always true that there is a semiprime between n and n^(0.3509). These conjectures are related to the various sequences about semiprime gaps and the merit of such gaps.
%C a(96) appears to be the last zero term. - _T. D. Noe_, Aug 12 2008
%H T. D. Noe, <a href="/A114021/b114021.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.
%e a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
%e a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
%e a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
%e a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...
%t SemiPrimeQ[n_] := TrueQ[Plus@@Last/@FactorInteger[n]==2]; Table[hi=n+Sqrt[n]; If[IntegerQ[hi], hi--, hi=Floor[hi]]; Length[Select[Range[n+1,hi], SemiPrimeQ]], {n,0,150}] (* _T. D. Noe_, Aug 12 2008 *)
%o (Perl) use ntheory ":all"; print "$_ ",semiprime_count($_+1, $_+sqrtint($_)-($_ && is_square($_))),"\n" for 0..1000; # _Dana Jacobsen_, Mar 04 2019
%Y Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.
%K easy,nonn
%O 0,9
%A _Jonathan Vos Post_, Jan 31 2006
%E Corrected and extended by _T. D. Noe_, Aug 12 2008
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