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

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, 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, 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

Offset: 0

Author

Jonathan Vos Post, Jan 31 2006

Keywords

• easy
• nonn

Comments

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.

a(96) appears to be the last zero term. - T. D. Noe, Aug 12 2008

Examples

			a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.

Programs

Mathematica

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 *)

Perl

use ntheory ":all"; print "$ ",semiprime_count($+1, $+sqrtint($)-($ && is_square($))),"\n" for 0..1000; # Dana Jacobsen, Mar 04 2019

Formula

a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.

Extensions

Corrected and extended by T. D. Noe, Aug 12 2008