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

1,4

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.

Links

Table of n, a(n) for n=1..57.

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 *)

Prog

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

Crossrefs

Cf. A046315, A205726, A205728.

Sequence in context: A321531 A194224 A194252 * A213609 A338237 A039823

Adjacent sequences: A205724 A205725 A205726 * A205728 A205729 A205730

Keyword

nonn

Author

Keith Backman, Jan 30 2012

Status

approved