A326497 - OEIS

A326497

Number of maximal sum-free and product-free subsets of {1..n}.

%I #20 Oct 28 2020 04:06:43

%S 1,1,1,1,2,3,4,6,8,9,15,21,26,38,51,69,89,119,149,197,261,356,447,601,

%T 781,1003,1293,1714,2228,2931,3697,4843,6258,8187,10273,13445,16894,

%U 21953,27469,35842,45410,58948,73939,95199,120593,154510,192995,247966,312642

%N Number of maximal sum-free and product-free subsets of {1..n}.

%C A set is sum-free and product-free if it contains no sum or product of two (not necessarily distinct) elements.

%H Fausto A. C. Cariboni, <a href="/A326497/b326497.txt">Table of n, a(n) for n = 0..68</a>

%H Andrew Howroyd, <a href="/A326497/a326497.txt">PARI Program</a>

%e The a(2) = 1 through a(10) = 15 subsets (A = 10):

%e {2} {23} {23} {23} {23} {237} {256} {267} {23A}

%e {34} {25} {256} {256} {258} {345} {345}

%e {345} {345} {267} {267} {357} {34A}

%e {456} {345} {345} {2378} {357}

%e {357} {357} {2569} {38A}

%e {4567} {2378} {2589} {2378}

%e {4567} {4567} {2569}

%e {5678} {4679} {2589}

%e {56789} {267A}

%e {269A}

%e {4567}

%e {4679}

%e {479A}

%e {56789}

%e {6789A}

%t fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];

%t Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Union[Plus@@@Tuples[#,2],Times@@@Tuples[#,2]]]=={}&]]],{n,0,10}]

%o (PARI) \\ See link for program file.

%o for(n=0, 37, print1(A326497(n), ", ")) \\ _Andrew Howroyd_, Aug 30 2019

%Y Sum-free and product-free subsets are A326495.

%Y Sum-free subsets are A007865.

%Y Maximal sum-free subsets are A121269.

%Y Product-free subsets are A326489.

%Y Maximal product-free subsets are A326496.

%Y Subsets with sums (and products) are A326083.

%Y Cf. A051026, A103580, A325710, A326076, A326117, A326491, A326492, A326498.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jul 09 2019

%E a(21)-a(40) from _Andrew Howroyd_, Aug 30 2019

%E a(41)-a(48) from _Jinyuan Wang_, Oct 11 2020