cp's OEIS Frontend

A114021 Number of semiprimes between n and n + sqrt(n).

%I A114021 #13 Mar 04 2019 18:47:29
%S A114021 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 A114021 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 A114021 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 A114021 Number of semiprimes between n and n + sqrt(n).
%C A114021 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 A114021 a(96) appears to be the last zero term. - _T. D. Noe_, Aug 12 2008
%H A114021 T. D. Noe, <a href="/A114021/b114021.txt">Table of n, a(n) for n = 0..10000</a>
%F A114021 a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.
%e A114021 a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
%e A114021 a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
%e A114021 a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
%e A114021 a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...
%t A114021 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 A114021 (Perl) use ntheory ":all"; print "$_ ",semiprime_count($_+1, $_+sqrtint($_)-($_ && is_square($_))),"\n" for 0..1000; # _Dana Jacobsen_, Mar 04 2019
%Y A114021 Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.
%K A114021 easy,nonn
%O A114021 0,9
%A A114021 _Jonathan Vos Post_, Jan 31 2006
%E A114021 Corrected and extended by _T. D. Noe_, Aug 12 2008