A046315 Odd semiprimes: odd numbers divisible by exactly 2 primes (counted with multiplicity).

9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 121, 123, 129, 133, 141, 143, 145, 155, 159, 161, 169, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 289

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Comments

In general, the prime factors, p, of a(n) are given by: p = sqrt(a(n) + (k/2)^2) +- (k/2) where k is the positive difference of the prime factors. Equivalently, p = (1/2)( sqrt(4a(n) + k^2) +- k ). - Wesley Ivan Hurt, Jun 28 2013

Examples

			From _K. D. Bajpai_, Jul 05 2014: (Start)
15 is a term because it is an odd number and 15 = 3 * 5, which is semiprime.
39 is a term because it is an odd number and 39 = 3 * 13, which is semiprime. (End)
		

Links

• Zak Seidov and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1956 terms from Zak Seidov)

Crossrefs

Odd members of A001358.

A046388 is a subsequence.

Cf. A085770 (number of odd semiprimes < 10^n). - Robert G. Wilson v, Aug 25 2011

Programs

Haskell

a046315 n = a046315_list !! (n-1)
a046315_list = filter odd a001358_list  -- Reinhard Zumkeller, Jan 02 2014

Maple

A046315 := proc(n) option remember; local r;
   if n = 1 then RETURN(9) fi;
   for r from procname(n - 1) + 2 by 2 do
      if numtheory[bigomega](r) = 2 then
         RETURN(r)
      end if
   end do
end proc:
seq(A046315(n),n=1..56); # Peter Luschny, Feb 15 2011

Mathematica

Reap[Do[If[Total[FactorInteger[n]][[2]] == 2, Sow[n]], {n, 1, 400, 2}]][[2,1]] (* Zak Seidov *)
fQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; Select[2 Range@ 150 - 1, fQ] (* Robert G. Wilson v, Feb 15 2011 *)
Select[Range[5,301,2],PrimeOmega[#]==2&] (* Harvey P. Dale, May 22 2015 *)

PARI

list(lim)=my(u=primes(primepi(lim\3)),v=List(),t); for(i=2,#u, for(j=i,#u, t=u[i]*u[j];if(t>lim,break); listput(v,t))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

Python

from math import isqrt
from sympy import primepi, primerange
def A046315(n):
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

Formula

Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 + P(2*s)) - P(s)/2^s, for s>1, where P is the prime zeta function. - Amiram Eldar, Nov 21 2020