|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
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)
|
|
|
LINKS
|
|
|
|
FORMULA
|
An integer is not an element for any integers i, j of the pairwise sum of {A001358(i)} and {A000290(j)}.
|
|
|
EXAMPLE
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
