A100570 - OEIS

%I #67 Dec 25 2020 11:02:08

%S 1,2,3,12,17,28,32,72,108,117,297,657

%N Numbers that are not the sum of a square and a semiprime.

%C No others up to 300000. Computed in collaboration with _Ray Chandler_. It appears that this sequence is finite, that is, that almost every positive integer is the sum of a semiprime and a square number. There are probably no further exceptions after a(12)=657.

%C The statement about the finiteness of this sequence (namely, a(n)<=657) is much stronger than the Goldbach binary conjecture. Indeed, a much weaker conjecture, that this sequence contains no perfect squares >1, already implies the Goldbach conjecture. Cf. comment in A241922. - _Vladimir Shevelev_, May 01 2014

%C From _Daniel Mikhail_, Nov 23 2020: (Start)

%C There are no new terms in this sequence between 658 and 2^28.

%C Notably, A014090 (numbers that are not the sum of a square and one prime) is a known infinite sequence. (End)

%H Daniel Mikhail, <a href="https://raw.githubusercontent.com/mikhaidn/SemiprimeCalculations/main/Summary%20of%202%5E28%20results">Brief glance of the search for integer solutions to a(n)-k^2 = semiprime</a>

%F An integer is not an element for any integers i, j of the pairwise sum of {A001358(i)} and {A000290(j)}.

%e From _Daniel Mikhail_, Nov 23 2020: (Start)

%e An integer m is in this set if, for any primes, p and q, there does not exist a natural k, such that m-k^2 = p*q.

%e Consider m=12 and all k such that k^2 < 12: k is either 0,1,4, or 9.

%e   12 - 0 = 12 = 2*2*2*3 => not semiprime;

%e   12 - 1 = 11 => not semiprime;

%e   12 - 4 = 8 = 2*2*2 => not semiprime;

%e   12 - 9 = 3 => not semiprime.

%e Therefore, 12 is a term. (End)

%t lim = 657; Complement[Range[lim],Select[Flatten[Outer[Plus,Select[Range[lim], PrimeOmega[#] == 2 &],Table[i^2, {i, 0, Sqrt[lim]}]]], # <= lim &]] (* _Robert Price_, Apr 10 2019 *)

%Y Cf. A000290, A001358, A046903, A241922, A014090.

%K nonn,hard,more

%O 1,2

%A _Jonathan Vos Post_, Nov 29 2004