A085770 Number of odd semiprimes < 10^n. Number of terms of A046315 < 10^n.

0, 1, 19, 204, 1956, 18245, 168497, 1555811, 14426124, 134432669, 1258822220, 11840335764, 111817881036, 1059796387004, 10076978543513, 96091983644261, 918679875630905, 8803388145953381, 84537081118605467, 813340036541900706, 7838825925851034479, 75669246175972479567

Offset

0,3

Links

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

Formula

a(n) = A066265(n) - A220262(n) for n > 1. - Jinyuan Wang, Jul 30 2021

Example

a(1)=1 because A046315(1)=9=3*3 is the only odd semiprime < 10^1,
a(2)=19 because there are 19 terms of A046315 < 10^2.

Mathematica

OddSemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, 2, PrimePi@ Sqrt@ n}]; Table[ OddSemiPrimePi[10^n], {n, 14}] (* Robert G. Wilson v, Feb 02 2006 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange
def A085770(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(3, s+1))) if n>1 else n # Chai Wah Wu, Oct 17 2024

Crossrefs

Cf. A046315 (odd numbers divisible by exactly 2 primes), A066265 (number of semiprimes < 10^n), A220262, A292785.

Sequence in context: A160452 A113596 A155670 * A002501 A180364 A125407

Adjacent sequences: A085767 A085768 A085769 * A085771 A085772 A085773

Keyword

nonn

Author

Hugo Pfoertner, Jul 22 2003

Extensions

a(10)-a(14) from Robert G. Wilson v, Feb 02 2006

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

a(0) inserted by and a(17)-a(21) from Jinyuan Wang, Jul 30 2021

Status

approved