A326081 - OEIS

%I #11 Aug 25 2019 19:44:25

%S 1,2,4,8,16,32,56,112,200,400,728,1456,2368,4736,8896,16112,30016,

%T 60032,105472,210944,366848,679680,1327232,2654464,4434176,8868352,

%U 17488640,33118336,60069248,120138496,206804224,413608448,759882880,1461600128,2909298496,5319739328

%N Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

%C For n > 0, this sequence divided by 2 first differs from A326116 at a(12)/2 = 1184, A326116(12) = 1232.

%C If A326117 counts product-free sets, this sequence counts product-closed sets.

%C The non-strict case is A326076.

%F For n > 0, a(n) = 2 * A308542(n).

%e The a(6) = 56 subsets:

%e   {}  {1}  {1,2}  {1,2,4}  {1,2,3,6}  {1,2,3,4,6}  {1,2,3,4,5,6}

%e       {2}  {1,3}  {1,2,5}  {1,2,4,5}  {1,2,3,5,6}

%e       {3}  {1,4}  {1,2,6}  {1,2,4,6}  {1,2,4,5,6}

%e       {4}  {1,5}  {1,3,4}  {1,2,5,6}  {1,3,4,5,6}

%e       {5}  {1,6}  {1,3,5}  {1,3,4,5}  {2,3,4,5,6}

%e       {6}  {2,4}  {1,3,6}  {1,3,4,6}

%e            {2,5}  {1,4,5}  {1,3,5,6}

%e            {2,6}  {1,4,6}  {1,4,5,6}

%e            {3,4}  {1,5,6}  {2,3,4,6}

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

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

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

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

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

%e                   {3,4,6}

%e                   {3,5,6}

%e                   {4,5,6}

%t Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[Times@@@Subsets[#,{2}],#<=n&]]&]],{n,0,10}]

%Y Cf. A007865, A051026, A103580, A196724, A308542, A326020, A326023, A326076, A326078, A326079.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 05 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 24 2019