A326496 Number of maximal product-free subsets of {1..n}.

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 6, 6, 9, 9, 15, 17, 30, 30, 46, 46, 51, 61, 103, 103, 129, 158, 282, 282, 322, 322, 553, 553, 615, 689, 1247, 1365, 1870, 1870, 3566, 3758, 5244, 5244, 8677, 8677, 9807, 12147, 23351, 23351, 27469, 31694, 45718, 47186, 54594, 54594, 95382, 108198

Offset: 0

Author

Gus Wiseman, Jul 09 2019

Keywords

• nonn

Comments

A set is product-free if it contains no product of two (not necessarily distinct) elements.

Also the number of maximal quotient-free subsets of {1..n}.

Examples

			The a(2) = 1 through a(10) = 6 subsets (A = 10):
  {2}  {23}  {23}  {235}  {235}   {2357}   {23578}   {23578}   {23578}
             {34}  {345}  {256}   {2567}   {25678}   {256789}  {2378A}
                          {3456}  {34567}  {345678}  {345678}  {256789}
                                                     {456789}  {26789A}
                                                               {345678A}
                                                               {456789A}
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..85
• P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.
• Andrew Howroyd, PARI Program

Crossrefs

Product-free subsets are A326489.

Subsets without products of distinct elements are A326117.

Maximal sum-free subsets are A121269.

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

Maximal subsets without products of distinct elements are A325710.

Cf. A007865, A051026, A326076, A326491, A326492, A326495, A327591.

Programs

Mathematica

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Times@@@Tuples[#,2]]=={}&]]],{n,0,10}]

PARI

\\ See link for program file.
for(n=0, 30, print1(A326496(n), ", ")) \\ Andrew Howroyd, Aug 30 2019

Extensions

a(18)-a(55) from Andrew Howroyd, Aug 30 2019