A100570 Numbers that are not the sum of a square and a semiprime.

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

Offset: 1

Author

Jonathan Vos Post, Nov 29 2004

Keywords

• nonn
• hard
• more

Comments

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.

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

From Daniel Mikhail, Nov 23 2020: (Start)

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

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

Examples

			From _Daniel Mikhail_, Nov 23 2020: (Start)
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.
Consider m=12 and all k such that k^2 < 12: k is either 0,1,4, or 9.
  12 - 0 = 12 = 2*2*2*3 => not semiprime;
  12 - 1 = 11 => not semiprime;
  12 - 4 = 8 = 2*2*2 => not semiprime;
  12 - 9 = 3 => not semiprime.
Therefore, 12 is a term. (End)
		

Links

• Daniel Mikhail, Brief glance of the search for integer solutions to a(n)-k^2 = semiprime

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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