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.

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

KEYWORD

nonn

AUTHOR

Keith Backman, Jan 30 2012

STATUS

approved