

A100570


Positive integers that cannot be partitioned into the sum of a semiprime and a square. Squares include 0 and 1.


8



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


LINKS

Table of n, a(n) for n=1..12.


FORMULA

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


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

Cf. A000290, A001358, A046903.
Sequence in context: A046486 A073452 A112976 * A056700 A299547 A140989
Adjacent sequences: A100567 A100568 A100569 * A100571 A100572 A100573


KEYWORD

nonn,more


AUTHOR

Jonathan Vos Post, Nov 29 2004


STATUS

approved



