A205728 -id:A205728 - cp's OEIS frontend

A205726 Number of semiprimes <= n^2.

0, 1, 3, 6, 9, 13, 17, 22, 26, 34, 40, 48, 56, 62, 75, 82, 90, 103, 114, 126, 135, 149, 164, 179, 190, 202, 220, 236, 253, 270, 289, 304, 320, 340, 360, 381, 404, 425, 443, 462, 484, 508, 533, 556, 581, 604, 634, 655, 678, 709, 738, 761, 783, 813, 846, 881

Offset: 1

Keith Backman, Jan 30 2012

nonn

See A205727 and A205728 for related sequences and relationship to Goldbach conjecture.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038108, A205727, A205728.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 100;  t = Select[Range[nn^2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
Module[{nn=60,sp},sp=Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,nn^2}]];Table[sp[[i^2]],{i,nn}]] (* Harvey P. Dale, May 29 2014 *)

from sympy import prime, primepi
def A205726(n): return int(sum(primepi(n**2//prime(k))-k+1 for k in range(1,primepi(n)+1))) # Chai Wah Wu, Jul 23 2024

a(n) = A072000(A000290(n)). - Michel Marcus, Sep 02 2013

A205727 Number of odd semiprimes <= n^2.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 8, 11, 14, 19, 23, 28, 33, 37, 46, 51, 56, 66, 73, 80, 88, 96, 108, 118, 126, 134, 148, 159, 172, 183, 197, 207, 220, 234, 249, 263, 280, 297, 309, 323, 338, 356, 376, 393, 412, 427, 449, 465, 482, 505, 527, 544, 561, 582, 606, 634, 658

Offset: 1

Keith Backman, Jan 30 2012

nonn

The Goldbach conjecture being true would imply that for every integer j, there exists at least one integer k such that (j^2)-(k^2) is an odd semiprime; i.e., for 2j=p+q, j=(p+q)/2 and k=(p-q)/2 results in (j^2)-(k^2)=pq. [Note that in many cases, 2j can be expressed as the sum of more than one set of two primes.] See A205728 for related series where p must be distinct from q.

Cf. A046315, A205726, A205728.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])];nn = 100;  t = Select[Range[1, nn^2, 2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
With[{osp=Table[{n,PrimeOmega[n]},{n,1,10001,2}]},Table[ Count[ Select[ osp,#[[1]]<=k^2&],?(#[[2]]==2&)],{k,60}]] (* _Harvey P. Dale, Dec 29 2017 *)

a(n) = sum(k=1, n^2, (k%2) && (bigomega(k) == 2)); \\ Michel Marcus, Feb 24 2018

cp's OEIS Frontend

A205726 Number of semiprimes <= n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula