A046315 - OEIS

%I #70 Aug 17 2024 01:34:20

%S 9,15,21,25,33,35,39,49,51,55,57,65,69,77,85,87,91,93,95,111,115,119,

%T 121,123,129,133,141,143,145,155,159,161,169,177,183,185,187,201,203,

%U 205,209,213,215,217,219,221,235,237,247,249,253,259,265,267,287,289

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

%C 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

%H Zak Seidov and K. D. Bajpai, <a href="/A046315/b046315.txt">Table of n, a(n) for n = 1..10000</a> (first 1956 terms from Zak Seidov)

%F 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

%e From _K. D. Bajpai_, Jul 05 2014: (Start)

%e 15 is a term because it is an odd number and 15 = 3 * 5, which is semiprime.

%e 39 is a term because it is an odd number and 39 = 3 * 13, which is semiprime. (End)

%p A046315 := proc(n) option remember; local r;

%p    if n = 1 then RETURN(9) fi;

%p    for r from procname(n - 1) + 2 by 2 do

%p       if numtheory[bigomega](r) = 2 then

%p          RETURN(r)

%p       end if

%p    end do

%p end proc:

%p seq(A046315(n),n=1..56); # _Peter Luschny_, Feb 15 2011

%t Reap[Do[If[Total[FactorInteger[n]][[2]] == 2, Sow[n]], {n, 1, 400, 2}]][[2,1]] (* _Zak Seidov_ *)

%t fQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; Select[2 Range@ 150 - 1, fQ] (* _Robert G. Wilson v_, Feb 15 2011 *)

%t Select[Range[5,301,2],PrimeOmega[#]==2&] (* _Harvey P. Dale_, May 22 2015 *)

%o (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

%o (Haskell)

%o a046315 n = a046315_list !! (n-1)

%o a046315_list = filter odd a001358_list  -- _Reinhard Zumkeller_, Jan 02 2014

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange

%o def A046315(n):

%o     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)))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Aug 17 2024

%Y Odd members of A001358.

%Y A046388 is a subsequence.

%Y Cf. A085770 (number of odd semiprimes < 10^n). - _Robert G. Wilson v_, Aug 25 2011

%K nonn

%O 1,1

%A _Patrick De Geest_, Jun 15 1998