A140114 Number of semiprimes strictly between n^2 and (n+1)^2.

0, 0, 1, 3, 2, 4, 3, 5, 4, 8, 5, 8, 7, 6, 13, 7, 7, 13, 10, 12, 9, 14, 14, 15, 11, 12, 18, 16, 16, 17, 18, 15, 16, 20, 20, 21, 22, 21, 18, 19, 21, 24, 24, 23, 25, 23, 29, 21, 23, 31, 29, 23, 21, 30, 33, 35, 34, 27, 30, 28, 29, 32, 30, 31, 36, 36, 36, 36, 36, 43, 24, 40, 38, 40, 39

Offset: 0

Author

Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008

Keywords

• nonn

Comments

Can it be proved that a(n)>0 for n>1?

Chen proves that there is a semiprime between n^2 and (n+1)^2 for sufficiently large n. - T. D. Noe, Oct 17 2008

Examples

			The first semiprimes are 6,10,14,15,21,22,26. None are <4, hence a(0)=a(1)=0.
One only is < 9, hence a(2) = 1.
Three more, 10, 14, 15 are < 16, hence a(3)=3.
		

References

• Jing Run Chen, On the distribution of almost primes in an interval, Sci. Sinica 18 (1975), 611-627.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A014085.

Programs

Mathematica

SemiPrimeQ[n_] := TrueQ[Plus@@Last/@FactorInteger[n]==2]; Table[Length[Select[Range[n^2+1, n^2+2n], SemiPrimeQ]], {n,0,100}] (* T. D. Noe, Sep 25 2008 *)
Module[{nn=80,sps},sps=Table[If[PrimeOmega[n]==2,1,0],{n,(nn+1)^2}];Table[ Total[ Take[sps,{k^2+1,(k+1)^2-1}]],{k,0,nn}]] (* Harvey P. Dale, Oct 03 2022 *)

PARI

a(n)=sum(k=n^2+1,n^2+2*n, bigomega(k)==2) \\ Charles R Greathouse IV, Jan 31 2017

Extensions

Corrected, edited and extended by T. D. Noe, Sep 25 2008