A085770 -id:A085770 - cp's OEIS frontend

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

Patrick De Geest, Jun 15 1998

nonn

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

			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)

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

Odd members of A001358.
A046388 is a subsequence.
Cf. A085770 (number of odd semiprimes < 10^n). - Robert G. Wilson v, Aug 25 2011

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

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

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 *)

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

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

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

A220262 Number of even semiprimes < 10^n. Number of terms of A100484 < 10^n.

Original entry on oeis.org

0, 3, 15, 95, 669, 5133, 41538, 348513, 3001134, 26355867, 234954223, 2119654578, 19308136142, 177291661649, 1638923764567, 15237833654620, 142377417196364, 1336094767763971, 12585956566571620, 118959989688273472, 1127779923790184543, 10720710117789005897

Offset: 0

Robert G. Wilson v, Dec 08 2012

nonn

All such semiprimes have the form 2*p, where p is prime. - T. D. Noe, Dec 09 2012

Kim Walisch, Fast C++ prime counting function implementation.

Cf. A066265, A085770, A100484.

Mathematica
```
Table[PrimePi[10^n/2], {n, 0, 14}]
```

a(n)=primepi(10^n\2) \\ Charles R Greathouse IV, Sep 08 2015

from sympy import primepi
def A220262(n): return primepi(10**n>>1) # Chai Wah Wu, Oct 17 2024

a(n) = A066265(n) - A085770(n) for n > 1.

a(15)-a(20) from Hugo Pfoertner, Oct 14 2017

a(21) from Jinyuan Wang, Jul 30 2021

A362042 Number of odd semiprimes less than 2^n.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 11, 24, 51, 103, 207, 417, 815, 1622, 3164, 6210, 12146, 23711, 46295, 90307, 176369, 344155, 672091, 1312721, 2565048, 5013566, 9804910, 19183069, 37547164, 73526846, 144042323, 282317826, 553564500, 1085869406, 2130916524, 4183381508, 8215884036

Offset: 0

Sidney Cadot, Apr 15 2023

nonn

Odd numbers with two prime factors are used as the modulus in the RSA algorithm. This sequence shows the growth of the number of 'candidate' RSA moduli for keys up to a given number of bits.

			For n=5, there are four integers less than 32 (i.e., 2^5) that are the product of two odd primes: {3*3, 3*5, 3*7, 5*5} = {9, 15, 21, 25}; hence, a(5)=4.

Cf. A046315, A007053, A085770, A125527.

a[n_]:=Length@Select[Range[1, 2^n - 1, 2], Total[Last /@ FactorInteger[#]] == 2 &]
Table[a[n],{n,0,24}]

a(n) = A125527(n) - A007053(n-1) for n > 0. - Jinyuan Wang, Apr 16 2023

More terms from Jinyuan Wang, Apr 16 2023

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

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A220262 Number of even semiprimes < 10^n. Number of terms of A100484 < 10^n.

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A362042 Number of odd semiprimes less than 2^n.

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