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A100570
Numbers that are not the sum of a square and a semiprime.
8
1, 2, 3, 12, 17, 28, 32, 72, 108, 117, 297, 657
OFFSET
1,2
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)
FORMULA
An integer is not an element for any integers i, j of the pairwise sum of {A001358(i)} and {A000290(j)}.
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Jonathan Vos Post, Nov 29 2004
STATUS
approved