A066265 a(n) = number of semiprimes < 10^n.

0, 3, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536, 1493776443, 13959990342, 131126017178, 1237088048653, 11715902308080, 111329817298881, 1061057292827269, 10139482913717352, 97123037685177087, 932300026230174178, 8966605849641219022, 86389956293761485464

Offset

0,2

Comments

Apart from the first nonzero term the sequence is identical to A036352. - Hugo Pfoertner, Jul 22 2003

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Semiprime

Index entries for sequences related to numbers of primes in various ranges

Formula

(1/2)*( pi(10^(n/2)) + Sum_{i=1..pi(10^n)} pi( (10^n-1)/P_i) ) = Sum_{i=1..pi(sqrt(10^n))} pi( (10^n-1)/P_i ) - binomial( pi(sqrt(10^n)), 2). - Robert G. Wilson v, May 16 2005

Example

Below 10 there are three semiprimes: 4 (2*2), 6 (2*3) and 9 (3*3).

Mathematica

f[n_] := Sum[ PrimePi[(10^n - 1)/Prime[i]], {i, PrimePi[ Sqrt[10^n]]}] - Binomial[ PrimePi[ Sqrt[10^n]], 2]; Do[ Print[ f[n]], {n, 0, 14}] (* Robert G. Wilson v, May 16 2005 *)
SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; Array[ SemiPrimePi[10^# - 1] &, 14, 0] (* Robert G. Wilson v, Jan 21 2015 *)

Prog

(PARI) a(n)=my(s); forprime(p=2, sqrt(10^n), s+=primepi((10^n-1)\p)); s-binomial(primepi(sqrt(10^n)), 2) \\ Charles R Greathouse IV, Apr 23 2012
(Perl) use Math::Prime::Util qw/:all/; use integer; sub countsp { my($k, $sum, $pc)=($_[0]-1, 0, 1); prime_precalc(60_000_000); forprimes { $sum += prime_count($k/$_) + 1 - $pc++; } int(sqrt($k)); $sum; } foreach my $n (0..16) { say "$n: ", countsp(10**$n); } # Dana Jacobsen, May 11 2014

Crossrefs

Cf. A001358, A064911, A072000, A036352 (identical starting from a(2)), A220262, A292785.

Sequence in context: A121077 A024396 A246384 * A268802 A284891 A231593

Adjacent sequences: A066262 A066263 A066264 * A066266 A066267 A066268

Keyword

nonn

Author

Patrick De Geest, Dec 10 2001

Extensions

More terms from Hugo Pfoertner, Jul 22 2003

a(14) from Robert G. Wilson v, May 16 2005

a(15)-a(16) from Donovan Johnson, Mar 18 2010

a(17)-a(18) from Dana Jacobsen, May 11 2014

a(19)-a(21) from Henri Lifchitz, Jul 04 2015

Status

approved