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

1,1

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

Links

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

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

Example

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)

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 *)

Prog

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

Crossrefs

Odd members of A001358.

A046388 is a subsequence.

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

Sequence in context: A175076 A046337 A359596 * A046372 A025045 A107987

Adjacent sequences: A046312 A046313 A046314 * A046316 A046317 A046318

Keyword

nonn

Author

Patrick De Geest, Jun 15 1998

Status

approved