A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

0, 1, 1, 4, 4, 12, 20, 40, 75, 147, 285, 535, 1062, 2006, 3918, 7548, 14595, 28293, 54761, 106452, 206421, 401522, 780966, 1520543, 2962226, 5777162, 11272279, 22009839, 43006972, 84077384, 164482781, 321944211, 630487562, 1235382703

Offset: 0

Author

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

Keywords

• nonn

Comments

The partial sum equals the number of Pi_2(2^n) = A125527(n).

Examples

			(2^2, 2^3] there is one semiprime, namely 6. 4 was counted in the previous entry.
		

Links

• Dana Jacobsen, Table of n, a(n) for n = 0..62 (first 48 terms from Charles R Greathouse IV, corrected a(47)-a(48))

Crossrefs

Cf. A001358, A072000, A066265, A036378, A120033-A120043.

Programs

Mathematica

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; t = Table[SemiPrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

PARI

pi2(n)=my(s,i); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
a(n)=pi2(2^(n+1))-pi2(2^n) \\ Charles R Greathouse IV, May 15 2012

Perl

use ntheory ":all"; print "$ ",semiprime_count(1+(1<<$), 1<<($+1)),"\n" for 0..48; # _Dana Jacobsen, Mar 04 2019

Perl

use ntheory ":all"; my $l=0; for (0..48) { my $c=semiprime_count(1<<($+1)); print "$ ",$c-$l,"\n"; $l=$c; } # Dana Jacobsen, Mar 04 2019