A120641 -id:A120641 - cp's OEIS frontend

A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n.

1, 2, 3, 5, 7, 12, 16, 27, 37, 58, 80, 131, 171, 277, 380, 580, 785, 1250, 1655, 2616, 3516, 5344, 7257, 11353, 14931, 23204, 31379, 47511, 63778, 98681, 130503, 201357, 270038, 407429, 548090, 840171, 1110429, 1701872, 2284325, 3440337, 4601656

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

The summands are allowed to be equal. The case where they must be distinct is A326080. If A007865 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).
From Gus Wiseman, Jul 08 2019: (Start)
Also the number of subsets of {1..n} containing no sum of any multiset of the elements. For example, the a(0) = 1 through a(6) = 16 subsets are:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {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}    {3,4}
                              {4,5}    {3,5}
                              {3,4,5}  {4,5}
                                       {4,6}
                                       {5,6}
                                       {3,4,5}
                                       {4,5,6}
(End)

			The a(0) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {2}    {2}      {3}        {3}          {4}
           {1,2}  {3}      {4}        {4}          {5}
                  {2,3}    {2,4}      {5}          {6}
                  {1,2,3}  {3,4}      {2,4}        {3,6}
                           {2,3,4}    {3,4}        {4,5}
                           {1,2,3,4}  {3,5}        {4,6}
                                      {4,5}        {5,6}
                                      {2,4,5}      {2,4,6}
                                      {3,4,5}      {3,4,6}
                                      {2,3,4,5}    {3,5,6}
                                      {1,2,3,4,5}  {4,5,6}
                                                   {2,4,5,6}
                                                   {3,4,5,6}
                                                   {2,3,4,5,6}
                                                   {1,2,3,4,5,6}
The a(7) = 27 subsets:
  {}  {4}  {36}  {246}  {2467}  {24567}  {234567}  {1234567}
      {5}  {45}  {356}  {3467}  {34567}
      {6}  {46}  {367}  {3567}
      {7}  {47}  {456}  {4567}
           {56}  {457}
           {57}  {467}
           {67}  {567}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326080.

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

For n > 0, a(n) = A103580(n) + 1.

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325

Offset: 0

Gus Wiseman, Aug 15 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(16) = 6 through a(22) = 12 strict partitions:
  (16)     (17)     (18)     (19)     (20)      (21)      (22)
  (9,7)    (9,8)    (10,8)   (10,9)   (11,9)    (12,9)    (13,9)
  (10,6)   (10,7)   (11,7)   (11,8)   (12,8)    (13,8)    (14,8)
  (11,5)   (11,6)   (13,5)   (12,7)   (13,7)    (15,6)    (15,7)
  (13,3)   (12,5)   (14,4)   (13,6)   (14,6)    (16,5)    (16,6)
  (7,5,4)  (13,4)   (7,6,5)  (14,5)   (17,3)    (17,4)    (17,5)
           (14,3)   (8,7,3)  (15,4)   (8,7,5)   (19,2)    (18,4)
           (15,2)            (16,3)   (9,6,5)   (11,10)   (19,3)
           (7,6,4)           (17,2)   (9,7,4)   (8,7,6)   (12,10)
                             (8,6,5)  (11,5,4)  (9,7,5)   (9,7,6)
                             (9,6,4)            (10,7,4)  (9,8,5)
                                                (10,8,3)  (7,6,5,4)
                                                (11,6,4)
                                                (11,7,3)

For sums of subsets instead of combinations of partitions we have A151897.
For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The complement in strict partitions is A364839, non-strict A364913.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A116861 and A364916 count linear combinations of strict partitions.
A323092 (ranks A320340) and A120641 count double-free partitions.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combs[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364350(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

More terms and offset corrected by Martin Fuller, Sep 11 2023

A326080 Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 31, 47, 73, 110, 168, 247, 375, 546, 817, 1193, 1769, 2552, 3791, 5445, 8012, 11517, 16899, 24144, 35391, 50427, 73614, 104984, 152656, 216802, 315689, 447473, 648813, 920163, 1332991, 1884735, 2728020, 3853437, 5568644, 7868096, 11347437

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.
The summands must be distinct. The case where they are allowed to be equal is A326083.
If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).

			The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {2}    {2}      {2}        {2}
           {1,2}  {3}      {3}        {3}
                  {1,3}    {4}        {4}
                  {2,3}    {1,4}      {5}
                  {1,2,3}  {2,3}      {1,5}
                           {2,4}      {2,4}
                           {3,4}      {2,5}
                           {1,3,4}    {3,4}
                           {2,3,4}    {3,5}
                           {1,2,3,4}  {4,5}
                                      {1,4,5}
                                      {2,3,5}
                                      {2,4,5}
                                      {3,4,5}
                                      {1,3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
The a(6) = 31 subsets:
  {}  {1}  {1,6}  {1,5,6}  {1,4,5,6}  {1,3,4,5,6}  {1,2,3,4,5,6}
      {2}  {2,5}  {2,3,5}  {2,3,5,6}  {2,3,4,5,6}
      {3}  {2,6}  {2,4,6}  {2,4,5,6}
      {4}  {3,4}  {2,5,6}  {3,4,5,6}
      {5}  {3,5}  {3,4,5}
      {6}  {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326083.

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

a(n)={
   my(recurse(k, b)=
      if( k > n, 1,
          my(t=self()(k + 1, b + (1<Andrew Howroyd, Aug 30 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019

A323092 Number of double-free integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 17, 24, 30, 40, 50, 66, 81, 104, 128, 161, 197, 246, 300, 369, 446, 546, 656, 796, 952, 1148, 1366, 1637, 1940, 2311, 2730, 3234, 3806, 4489, 5262, 6181, 7225, 8454, 9846, 11484, 13335, 15499, 17948, 20796, 24017, 27751, 31970, 36837

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

An integer partition is double-free if no part is twice any other part.

			The a(1) = 1 through a(8) = 14 double-free integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (431)
                                               (4111)     (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A051424, A101417, A120641, A276431, A305148, A323093, A323094.

Table[Length[Select[IntegerPartitions[n],Intersection[#,2*#]=={}&]],{n,30}]

A320340 Heinz numbers of double-free integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 07 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is double-free if no part is twice any other part.
Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

A050291 Number of double-free subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 288, 576, 960, 1920, 2880, 5760, 9360, 18720, 28080, 56160, 93600, 187200, 280800, 561600, 898560, 1797120, 2695680, 5391360, 8985600, 17971200, 26956800, 53913600, 87091200, 174182400, 261273600, 522547200, 870912000

Offset: 0

Eric W. Weisstein

A set is double-free if it does not contain both x and 2x.

So these are equally "half-free" subsets. - Gus Wiseman, Jul 08 2019

			From _Gus Wiseman_, Jul 08 2019: (Start)
The a(0) = 1 through a(5) = 20 double-free subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                {2,3}  {1,3}    {5}
                       {1,4}    {1,3}
                       {2,3}    {1,4}
                       {3,4}    {1,5}
                       {1,3,4}  {2,3}
                                {2,5}
                                {3,4}
                                {3,5}
                                {4,5}
                                {1,3,4}
                                {1,3,5}
                                {1,4,5}
                                {2,3,5}
                                {3,4,5}
                                {1,3,4,5}
(End)

Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 1989.

Alois P. Heinz, Table of n, a(n) for n = 0..4030 (terms n = 1..400 from T. D. Noe)
Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]
Eric Weisstein's World of Mathematics, Double-Free Set.

Cf. A000045, A007814, A050292-A050296.

Cf. A007865, A103580, A120641, A308546, A320340, A323092, A326083, A326115.

a:= proc(n) option remember; `if`(n=0, 1, (F-> (p-> a(n-1)*F(p+3)
      /F(p+2))(padic[ordp](n, 2)))(j-> (<<0|1>, <1|1>>^j)[1, 2]))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 16 2019

a[n_] := a[n] = (b = IntegerExponent[2n, 2]; a[n-1]*Fibonacci[b+2]/Fibonacci[b+1]); a[1]=2; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Oct 10 2012, from first formula *)
Table[Length[Select[Subsets[Range[n]],Intersection[#,#/2]=={}&]],{n,0,10}] (* Gus Wiseman, Jul 08 2019 *)

first(n)=my(v=vector(n)); v[1]=2; for(k=2,n, v[k]=v[k-1]*fibonacci(valuation(k,2)+3)/fibonacci(valuation(k,2)+2)); v \\ Charles R Greathouse IV, Feb 07 2017

a(n) = a(n-1)*Fibonacci(b(2n)+2)/Fibonacci(b(2n)+1), Fibonacci = A000045, b = A007814.

a(n) = 2^n - A088808(n). - Reinhard Zumkeller, Oct 19 2003

Extended with formula by Christian G. Bower, Sep 15 1999

a(0)=1 prepended by Alois P. Heinz, Jan 16 2019

A350842 Number of integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Heinz number rankings are in parentheses below.
The version for no difference 0 is A000009.
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The version for all differences > -2 is A034296, strict A001227.
The opposite version is A072670.
The version for no difference -1 is A116931 (A319630), strict A003114.
The multiplicative version is A350837 (A350838), strict A350840.
The strict case is A350844.
The complement for quotients is counted by A350846 (A350845).
A000041 = integer partitions.
A027187 = partitions of even length.
A027193 = partitions of odd length (A026424).
A323092 = double-free partitions (A320340), strict A120641.
A325534 = separable partitions (A335433).
A325535 = inseparable partitions (A335448).
A350839 = partitions with a gap and conjugate gap (A350841).
Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

A308546 Number of double-closed subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 60, 120, 180, 360, 480, 960, 1440, 2880, 3456, 6912, 10368, 20736, 27648, 55296, 82944, 165888, 207360, 414720, 622080, 1244160, 1658880, 3317760, 4976640, 9953280, 11612160, 23224320, 34836480, 69672960, 92897280

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

These are subsets containing twice any element whose double is <= n.
Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:
  {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
      {3}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
      {5}  {1,5}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
           {3,5}  {1,3,5}  {1,2,4,5}
           {3,6}  {1,3,6}  {1,3,5,6}
                  {3,5,6}

			The a(6) = 24 subsets:
  {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
      {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
           {4,6}  {3,4,6}  {2,4,5,6}
           {5,6}  {3,5,6}  {3,4,5,6}
                  {4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..500

Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

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

From Charlie Neder, Jun 10 2019: (Start)
a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).
a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

a(21)-a(36) from Charlie Neder, Jun 10 2019

A350844 Number of strict integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset: 0

Gus Wiseman, Jan 21 2022

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for no difference 0 is A000009.
The version for no difference > -2 is A001227, non-strict A034296.
The version for no difference -1 is A003114 (A325160).
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The opposite version is A072670.
The multiplicative version is A350840, non-strict A350837 (A350838).
The non-strict version is A350842.
A000041 counts integer partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length (A026424).
A116931 counts partitions with no difference -1 (A319630).
A323092 counts double-free integer partitions (A320340) strict A120641.
A325534 counts separable partitions (A335433).
A325535 counts inseparable partitions (A335448).
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],0|-2]&]],{n,0,30}]

A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 9, 12, 13, 18, 23, 29, 37, 49, 55, 71, 84, 104, 126, 153, 185, 221, 261, 317, 375, 446, 523, 623, 721, 854, 994, 1168, 1357, 1579, 1833, 2126, 2455, 2843, 3270, 3766, 4320, 4980, 5687, 6521, 7444, 8498, 9684, 11039, 12540, 14262

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(1) = 1 through a(8) = 12 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (311)    (51)      (52)       (53)
                    (1111)  (11111)  (222)     (61)       (62)
                                     (3111)    (322)      (71)
                                     (111111)  (331)      (332)
                                               (511)      (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

Cf. A018819, A101417, A120641, A276431, A305148, A323092, A323094.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]

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