A326117 -id:A326117 - cp's OEIS frontend

A151897 Number of subsets of {1, 2, ..., n} such that no member is a sum of distinct other members.

1, 2, 4, 7, 13, 22, 37, 60, 100, 155, 249, 381, 591, 889, 1365, 2009, 3047, 4453, 6602, 9567, 14151, 20228, 29654, 42302, 61369, 87108, 126066, 177580, 256039, 360304, 515740, 724069, 1036860, 1448746, 2069526, 2893311, 4117725, 5749540, 8186555

Offset: 0

David Wasserman, Apr 16 2008

nonn

This sequence and A085489 first differ at n = 7. a(7) = 60, A085489(7) = 61. A085489(7) includes {1, 2, 4, 7}, which is excluded from a(7) because 1+2+4 = 7.

If this sequence counts sum-free sets, A326080 counts sum-closed sets, which are different from sum-full sets (A093971). - Gus Wiseman, Jun 07 2019

			a(4) = 13, including all subsets of {1, 2, 3, 4} except {1, 2, 3} (excluded
because 1+2 = 3), {1, 3, 4} (excluded because 1+3 = 4), and {1, 2, 3, 4} (excluded for both reasons.)
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..85

Cf. A007865, A050291, A085489, A103580, A308546, A326080, A326083, A326117.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Plus@@@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,10}] (* Gus Wiseman, Jun 07 2019 *)

a(0) = 1 prepended by Gus Wiseman, Jun 07 2019

A085489 a(n) is the number of subsets of {1,...,n} containing no solutions to x+y=z with x and y distinct (one version of "sum-free subsets").

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 37, 61, 102, 162, 261, 410, 646, 1001, 1553, 2370, 3645, 5515, 8303, 12470, 18713, 27811, 41244, 60962, 89733, 131870, 192522, 281125, 408680, 593880, 855661, 1238592, 1779614, 2563476, 3660084, 5255913, 7473380, 10696444, 15137517

Offset: 0

Eric W. Weisstein, Jul 02 2003

First differs from A151897 at a(7) = 61, A151897(7) = 60. The one subset counted under a(7) but not under A151897(7) is {1,2,4,7}. - Gus Wiseman, Jun 07 2019

			From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}
The a(5) = 22 subsets:
  {}  {1}  {1,2}  {1,2,4}
      {2}  {1,3}  {1,2,5}
      {3}  {1,4}  {1,3,5}
      {4}  {1,5}  {2,3,4}
      {5}  {2,3}  {2,4,5}
           {2,4}  {3,4,5}
           {2,5}
           {3,4}
           {3,5}
           {4,5}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..75, (terms up to a(57) from Ben Burns)
Eric Weisstein's World of Mathematics, Sum-Free Set [Strictly speaking this link is not relevant, since it uses a different definition of "sum-free".]

See A007865 for another version.

Cf. A050291, A051026, A103580, A151897, A308546, A326080, A326083, A326117.

Table[Length[Select[Subsets[Range[n]],Intersection[ #,Select[ Plus@@@ Subsets[ #,{2}],#<=n&]]=={}&]],{n,0,10}] (* Gus Wiseman, Jun 07 2019 *)

a(n) = 2^n - A088809(n). - Reinhard Zumkeller, Oct 19 2003

More terms from Reinhard Zumkeller, Jul 13 2003
Edited by David Wasserman, Apr 16 2008
a(0) = 1 prepended by Gus Wiseman, Jun 07 2019

A051026 Number of primitive subsequences of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, 505, 733, 1133, 1529, 3057, 3897, 7793, 10241, 16513, 24593, 49185, 59265, 109297, 163369, 262489, 355729, 711457, 879937, 1759873, 2360641, 3908545, 5858113, 10534337, 12701537, 25403073, 38090337, 63299265, 81044097, 162088193, 205482593, 410965185, 570487233, 855676353

Offset: 0

Eric W. Weisstein

a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011

Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019

			a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,3}      {1,3}
                  {1,3}    {1,4}      {1,4}
                  {1,2,3}  {1,2,4}    {1,5}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {2}    {2}      {3}        {3}
           {1,2}  {3}      {4}        {4}
                  {2,3}    {2,4}      {5}
                  {1,2,3}  {3,4}      {2,4}
                           {2,3,4}    {3,4}
                           {1,2,3,4}  {3,5}
                                      {4,5}
                                      {2,3,4}
                                      {2,4,5}
                                      {3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
(End)
From _Gus Wiseman_, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,5}
                                      {1,3,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}
(End)

Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012

Juliana Couras, Ricardo Jesus, and Tomás Oliveira e Silva, Table of n, a(n) for n = 0..800 (terms up to n=80 from Alois P. Heinz)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Nathan McNew, Counting primitive subsets and other statistics of the divisor graph of {1,2,..,n}, arXiv:1808.04923 [math.NT], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Eric Weisstein's World of Mathematics, Primitive Sequence

Cf. A007865, A054519, A096827, A103580, A303362, A305148.
Cf. A326023, A326076, A326077, A326081, A326082, A326083, A326117.
Cf. A037031, A051026, A355740, A368110, A370585.

with(numtheory):
b:= proc(s) option remember; local n;
      n:= max(s[]);
      `if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
    end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011

b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Select[Union@@Table[#*i,{i,n}],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]],{n,0,10}] (* Gus Wiseman, Mar 12 2024 *)

More terms from David Wasserman, May 02 2002

a(32)-a(37) from Donovan Johnson, Aug 11 2010

A326489 Number of product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 44, 88, 136, 252, 504, 896, 1792, 3392, 6352, 9720, 19440, 35664, 71328, 129952, 247232, 477664, 955328, 1700416, 2657280, 5184000, 10368000, 19407360, 38814720, 68868352, 137736704, 260693504, 505830400, 999641600, 1882820608, 2807196672

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

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

			The a(0) = 1 through a(6) = 22 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {6}
                             {2,5}    {2,3}
                             {3,4}    {2,5}
                             {3,5}    {2,6}
                             {4,5}    {3,4}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,4,6}
                                      {3,5,6}
                                      {4,5,6}
                                      {3,4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..167, (terms up to a(100) from Andrew Howroyd)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Andrew Howroyd, PARI Program

Product-closed subsets are A326076.
Subsets containing no products are A326114.
Subsets containing no products of distinct elements are A326117.
Subsets containing no quotients are A327591.
Maximal product-free subsets are A326496.
Cf. A007865, A051026, A326023, A326081, A326116, A326495.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Times@@@Tuples[#,2]]=={}&]],{n,10}]

a(21)-a(36) from Andrew Howroyd, Aug 25 2019

a(0)=1 prepended to data, example and b-file by Peter Kagey, Sep 18 2019

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

Original entry on oeis.org

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

Gus Wiseman, Jul 09 2019

nonn

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}.

			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}

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

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.

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

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

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

A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 56, 112, 200, 400, 728, 1456, 2368, 4736, 8896, 16112, 30016, 60032, 105472, 210944, 366848, 679680, 1327232, 2654464, 4434176, 8868352, 17488640, 33118336, 60069248, 120138496, 206804224, 413608448, 759882880, 1461600128, 2909298496, 5319739328

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

For n > 0, this sequence divided by 2 first differs from A326116 at a(12)/2 = 1184, A326116(12) = 1232.
If A326117 counts product-free sets, this sequence counts product-closed sets.
The non-strict case is A326076.

			The a(6) = 56 subsets:
  {}  {1}  {1,2}  {1,2,4}  {1,2,3,6}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {2}  {1,3}  {1,2,5}  {1,2,4,5}  {1,2,3,5,6}
      {3}  {1,4}  {1,2,6}  {1,2,4,6}  {1,2,4,5,6}
      {4}  {1,5}  {1,3,4}  {1,2,5,6}  {1,3,4,5,6}
      {5}  {1,6}  {1,3,5}  {1,3,4,5}  {2,3,4,5,6}
      {6}  {2,4}  {1,3,6}  {1,3,4,6}
           {2,5}  {1,4,5}  {1,3,5,6}
           {2,6}  {1,4,6}  {1,4,5,6}
           {3,4}  {1,5,6}  {2,3,4,6}
           {3,5}  {2,3,6}  {2,3,5,6}
           {3,6}  {2,4,5}  {2,4,5,6}
           {4,5}  {2,4,6}  {3,4,5,6}
           {4,6}  {2,5,6}
           {5,6}  {3,4,5}
                  {3,4,6}
                  {3,5,6}
                  {4,5,6}

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

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

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

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A326495 Number of subsets of {1..n} containing no sums or products of pairs of elements.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 30, 45, 71, 101, 171, 258, 427, 606, 988, 1328, 2141, 3116, 4952, 6955, 11031, 15320, 23978, 33379, 48698, 66848, 104852, 144711, 220757, 304132, 461579, 636555, 973842, 1316512, 1958827, 2585432, 3882842, 5237092, 7884276, 10555738, 15729292

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

The pairs are not required to be strict.

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}     {}     {}       {}
      {2}  {2}    {2}    {2}      {2}
           {3}    {3}    {3}      {3}
           {2,3}  {4}    {4}      {4}
                  {2,3}  {5}      {5}
                  {3,4}  {2,3}    {6}
                         {2,5}    {2,3}
                         {3,4}    {2,5}
                         {3,5}    {2,6}
                         {4,5}    {3,4}
                         {3,4,5}  {3,5}
                                  {4,5}
                                  {4,6}
                                  {5,6}
                                  {2,5,6}
                                  {3,4,5}
                                  {4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..90

Subsets without sums are A007865.
Subsets without products are A326489.
Subsets without differences or quotients are A326490.
Maximal subsets without sums or products are A326497.
Subsets with sums (and products) are A326083.
Cf. A051026, A103580, A326020, A326076, A326117, A326491.

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

For n > 0, a(n) = A326490(n) - 1.

a(19)-a(41) from Andrew Howroyd, Aug 25 2019

A326497 Number of maximal sum-free and product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 15, 21, 26, 38, 51, 69, 89, 119, 149, 197, 261, 356, 447, 601, 781, 1003, 1293, 1714, 2228, 2931, 3697, 4843, 6258, 8187, 10273, 13445, 16894, 21953, 27469, 35842, 45410, 58948, 73939, 95199, 120593, 154510, 192995, 247966, 312642

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

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

			The a(2) = 1 through a(10) = 15 subsets (A = 10):
  {2}  {23}  {23}  {23}   {23}   {237}   {256}   {267}    {23A}
             {34}  {25}   {256}  {256}   {258}   {345}    {345}
                   {345}  {345}  {267}   {267}   {357}    {34A}
                          {456}  {345}   {345}   {2378}   {357}
                                 {357}   {357}   {2569}   {38A}
                                 {4567}  {2378}  {2589}   {2378}
                                         {4567}  {4567}   {2569}
                                         {5678}  {4679}   {2589}
                                                 {56789}  {267A}
                                                          {269A}
                                                          {4567}
                                                          {4679}
                                                          {479A}
                                                          {56789}
                                                          {6789A}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..68
Andrew Howroyd, PARI Program

Sum-free and product-free subsets are A326495.
Sum-free subsets are A007865.
Maximal sum-free subsets are A121269.
Product-free subsets are A326489.
Maximal product-free subsets are A326496.
Subsets with sums (and products) are A326083.
Cf. A051026, A103580, A325710, A326076, A326117, A326491, A326492, A326498.

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

\\ See link for program file.
for(n=0, 37, print1(A326497(n), ", ")) \\ Andrew Howroyd, Aug 30 2019

a(21)-a(40) from Andrew Howroyd, Aug 30 2019

a(41)-a(48) from Jinyuan Wang, Oct 11 2020

A326116 Number of subsets of {2..n} containing no products of two or more distinct elements.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 100, 200, 364, 728, 1232, 2464, 4592, 8296, 15920, 31840, 55952, 111904, 195712, 362336, 697360, 1394720, 2334112, 4668224, 9095392, 17225312, 31242784, 62485568, 106668608, 213337216, 392606528, 755131840, 1491146912, 2727555424, 4947175712

Offset: 1

Gus Wiseman, Jun 06 2019

nonn

First differs from A308542 at a(12) = 1232, A308542(12) = 1184.

If this sequence counts product-free sets, A308542 counts product-closed sets.

			The a(6) = 28 subsets:
  {}  {2}  {2,3}  {2,3,4}  {2,3,4,5}
      {3}  {2,4}  {2,3,5}  {2,4,5,6}
      {4}  {2,5}  {2,4,5}  {3,4,5,6}
      {5}  {2,6}  {2,4,6}
      {6}  {3,4}  {2,5,6}
           {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..47
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.

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326117, A308542.

Table[Length[Select[Subsets[Range[2,n]],Intersection[#,Select[Times@@@Subsets[#,{2}],#<=n&]]=={}&]],{n,10}]

a(n)={
   my(recurse(k, ep)=
    if(k > n, 1,
      my(t = self()(k + 1, ep));
      if(!bittest(ep,k),
         forstep(i=n\k, 1, -1, if(bittest(ep,i), ep=bitor(ep,1<<(k*i))));
         t += self()(k + 1, ep);
      );
      t);
   );
   recurse(2, 2);
} \\ Andrew Howroyd, Aug 25 2019

For n > 0, a(n) = A326117(n) - 1.

Terms a(21)-a(36) from Andrew Howroyd, Aug 25 2019

A326491 Number of maximal subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 10, 16, 22, 27, 39, 52, 70, 90, 120, 150, 198, 262, 357, 448, 602, 782, 1004, 1294, 1715, 2229, 2932, 3698, 4844, 6259, 8188, 10274, 13446, 16895, 21954, 27470, 35843, 45411, 58949, 73940, 95200, 120594, 154511, 192996, 247967, 312643

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(1) = 1 through a(9) = 10 subsets:
  {1}  {1}  {1}    {1}    {1}      {1}      {1}        {1}        {1}
       {2}  {2,3}  {2,3}  {2,3}    {2,3}    {2,3,7}    {2,5,6}    {2,6,7}
                   {3,4}  {2,5}    {2,5,6}  {2,5,6}    {2,5,8}    {3,4,5}
                          {3,4,5}  {3,4,5}  {2,6,7}    {2,6,7}    {3,5,7}
                                   {4,5,6}  {3,4,5}    {3,4,5}    {2,3,7,8}
                                            {3,5,7}    {3,5,7}    {2,5,6,9}
                                            {4,5,6,7}  {2,3,7,8}  {2,5,8,9}
                                                       {4,5,6,7}  {4,5,6,7}
                                                       {5,6,7,8}  {4,6,7,9}
                                                                  {5,6,7,8,9}

Subsets without differences or quotients are A326490.
Subsets with differences and quotients are A326494.
Maximal subsets without differences are A121269
Maximal subsets without quotients are A326492.
Cf. A007865, A051026, A054519, A326023, A326117, A326489, A326496, A327591.

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

a(n) = A326497(n) + 1 for n > 1. - Andrew Howroyd, Aug 30 2019

a(16)-a(40) from Andrew Howroyd, Aug 30 2019

a(41)-a(48) from Jinyuan Wang, Mar 04 2025

cp's OEIS Frontend

A151897 Number of subsets of {1, 2, ..., n} such that no member is a sum of distinct other members.

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A085489 a(n) is the number of subsets of {1,...,n} containing no solutions to x+y=z with x and y distinct (one version of "sum-free subsets").

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A051026 Number of primitive subsequences of {1, 2, ..., n}.

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A326489 Number of product-free subsets of {1..n}.

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

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A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

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A326495 Number of subsets of {1..n} containing no sums or products of pairs of elements.

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