A326025 - OEIS

%I #13 Oct 05 2020 05:08:41

%S 1,1,2,2,2,4,5,10,13,20,28,40,54,82,120,172,244,347,471,651,874,1198,

%T 1635,2210,2867,3895,5234,6889,9019,11919,15629,20460,26254,33827,

%U 43881,56367,71841,91834,117695,148503,188039,311442,390859,488327,610685,759665

%N Number of maximal subsets of {1..n} containing no sums or products of distinct elements.

%H Andrew Howroyd, <a href="/A326025/a326025.txt">PARI Program</a>

%e The a(1) = 1 through a(8) = 13 maximal subsets:

%e   {1}  {1}  {1}    {1}      {1}      {1}        {1}        {1}

%e        {2}  {2,3}  {2,3,4}  {2,3,4}  {2,3,4}    {2,3,4}    {2,3,4}

%e                             {2,4,5}  {2,4,5}    {2,3,7}    {2,4,5}

%e                             {3,4,5}  {2,5,6}    {2,4,5}    {2,4,7}

%e                                      {3,4,5,6}  {2,4,7}    {2,5,6}

%e                                                 {2,5,6}    {2,5,8}

%e                                                 {2,6,7}    {2,6,7}

%e                                                 {3,4,5,6}  {2,3,7,8}

%e                                                 {3,5,6,7}  {3,4,5,6}

%e                                                 {4,5,6,7}  {3,4,6,8}

%e                                                            {3,5,6,7}

%e                                                            {3,6,7,8}

%e                                                            {4,5,6,7,8}

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

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

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

%o for(n=0, 25, print1(A326025(n), ", ")) \\ _Andrew Howroyd_, Aug 29 2019

%Y Maximal subsets without sums of distinct elements are A326498.

%Y Maximal subsets without products of distinct elements are A325710.

%Y Subsets without sums or products of distinct elements are A326024.

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

%Y Maximal sum-free and product-free subsets are A326497.

%Y Cf. A007865, A051026, A121269, A151897, A326116, A326117, A326491, A326495, A326496.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 09 2019

%E a(16)-a(40) from _Andrew Howroyd_, Aug 29 2019

%E a(41)-a(45) from _Jinyuan Wang_, Oct 03 2020