A205727 Number of odd semiprimes <= n^2.

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

Author

Keith Backman, Jan 30 2012

Keywords

• nonn

Comments

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.

Crossrefs

Cf. A046315, A205726, A205728.

Programs

Mathematica

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 *)

PARI

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