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 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 (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 LINKS 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

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Last modified October 16 13:32 EDT 2019. Contains 328093 sequences. (Running on oeis4.)