A108917 -id:A108917 - cp's OEIS frontend

A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367225 /\ A005117.

			The a(2) = 1 through a(11) = 7 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (10)     (11)
            (3,1)  (4,1)  (5,1)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                                 (6,1)  (7,1)  (6,3)  (7,3)    (7,4)
                                               (8,1)  (9,1)    (8,3)
                                                      (5,4,1)  (10,1)
                                                               (5,4,2)
                                                               (6,4,1)
The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15):
  2  3  4   5   6   7   8   9   A    B    C    D    E     F
        31  41  51  43  53  54  64   65   75   76   86    87
                    61  71  63  73   74   84   85   95    96
                            81  91   83   93   94   A4    A5
                                541  A1   B1   A3   B3    B4
                                     542  642  C1   D1    C3
                                     641  651  652  752   E1
                                          741  742  761   654
                                               751  842   762
                                               841  851   852
                                                    941   861
                                                    6521  942
                                                          951
                                                          A41
                                                          7521

Chai Wah Wu, Table of n, a(n) for n = 0..118

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215*   A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A007865/A085489/A151897 count certain types of sum-free subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A237667 counts sum-free partitions, ranks A364531.
A240861 counts strict partitions with length not a part, complement A240855.
A275972 counts strict knapsack partitions, non-strict A108917.
A364349 counts sum-free strict partitions, sum-full A364272.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365663 counts strict partitions without a subset-sum k, non-strict A046663.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Apr 07 2024

nonn

These partitions are counted by A371795, even case A006827.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 aerated and ranked by A357976.

			The prime indices of 975 are {2,3,3,6}, which are not biquanimous, so 975 is in the sequence.
The prime indices of 900 are {1,1,2,2,3,3}, which can be partitioned into {{1,2,3},{1,2,3}} or {{3,3},{1,1,2,2}}, so 900 is not in the sequence.

The complement is A357976, counted by A002219.
For prime signature instead of indices we have A371782, complement A371781.
Partitions of this type are counted by A371795, even case A006827.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A002221, A027187, A035470, A064914, A299701, A300061, A318434, A321455, A326534, A357879, A366320, A367094.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Select[Range[100],Not@*biqQ@*prix]

Numbers n without a divisor d|n such that A056239(d) = A056239(n/d).

A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 40, 41, 48, 50, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 144, 145, 160, 161, 176, 192, 194, 196, 208, 256, 257, 258, 260, 261, 262, 264, 265, 268, 272, 274

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

Also numbers whose binary indices together with 0 define a Golomb ruler.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The list of terms together with the corresponding compositions begins:
    0: ()        41: (2,3,1)    130: (6,2)      262: (6,1,2)
    1: (1)       48: (1,5)      132: (5,3)      264: (5,4)
    2: (2)       50: (1,3,2)    133: (5,2,1)    265: (5,3,1)
    4: (3)       64: (7)        134: (5,1,2)    268: (5,1,3)
    5: (2,1)     65: (6,1)      144: (3,5)      272: (4,5)
    6: (1,2)     66: (5,2)      145: (3,4,1)    274: (4,3,2)
    8: (4)       68: (4,3)      160: (2,6)      276: (4,2,3)
    9: (3,1)     69: (4,2,1)    161: (2,5,1)    288: (3,6)
   12: (1,3)     70: (4,1,2)    176: (2,1,5)    289: (3,5,1)
   16: (5)       72: (3,4)      192: (1,7)      290: (3,4,2)
   17: (4,1)     80: (2,5)      194: (1,5,2)    296: (3,2,4)
   18: (3,2)     81: (2,4,1)    196: (1,4,3)    304: (3,1,5)
   20: (2,3)     88: (2,1,4)    208: (1,2,5)    320: (2,7)
   24: (1,4)     96: (1,6)      256: (9)        321: (2,6,1)
   32: (6)       98: (1,4,2)    257: (8,1)      324: (2,4,3)
   33: (5,1)    104: (1,2,4)    258: (7,2)      328: (2,3,4)
   34: (4,2)    128: (8)        260: (6,3)      352: (2,1,6)
   40: (2,4)    129: (7,1)      261: (6,2,1)    384: (1,8)

Wikipedia, Golomb ruler

A subset of A233564.
Also a subset of A333223.
These compositions are counted by A169942 and A325677.
The number of distinct nonzero subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Lengths of optimal Golomb rulers are A003022.
Inequivalent optimal Golomb rulers are counted by A036501.
Complete rulers are A103295, with perfect case A103300.
Knapsack partitions are counted by A108917, with strict case A275972.
Distinct subsequences are counted by A124770 and A124771.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Knapsack compositions are counted by A325676.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A029931, A048793, A066099, A070939, A228351, A295235, A325678, A325680, A325768, A325779, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],UnsameQ@@ReplaceList[stc[#],{_,s__,_}:>Plus[s]]&]

A371796 Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 19, 43, 94, 206, 439, 946, 1990, 4204, 8761, 18233, 37778, 78151, 160296, 328670, 670193, 1363543, 2772436, 5632801, 11404156, 23071507, 46613529, 94098106, 189959349, 383407198, 773009751

Offset: 0

Gus Wiseman, Apr 17 2024

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is counted under a(9).
The a(3) = 1 through a(6) = 19 subsets:
  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
           {1,3,4}    {1,3,4}      {1,3,4}
           {1,2,3,4}  {1,4,5}      {1,4,5}
                      {2,3,5}      {1,5,6}
                      {1,2,3,4}    {2,3,5}
                      {1,2,4,5}    {2,4,6}
                      {2,3,4,5}    {1,2,3,4}
                      {1,2,3,4,5}  {1,2,3,6}
                                   {1,2,4,5}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {2,3,4,5}
                                   {2,3,5,6}
                                   {3,4,5,6}
                                   {1,2,3,4,5}
                                   {1,2,3,4,6}
                                   {1,2,4,5,6}
                                   {2,3,4,5,6}
                                   {1,2,3,4,5,6}

The "bi-" version for integer partitions is A002219 aerated, ranks A357976.
The "bi-" version for strict partitions is A237258 aerated, ranks A357854.
The complement for integer partitions is A321451, ranks A321453.
The version for integer partitions is A321452, ranks A321454
The version for strict partitions is A371737, complement A371736.
The complement is counted by A371789, differences A371790.
The "bi-" version is A371791, complement A371792.
First differences are A371797.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
Cf. A000005, A018818, A035470, A038041, A232466, A279791, A321142, A365661, A371731, A371794, A371795.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,0,10}]

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

Original entry on oeis.org

0, 1, 3, 6, 12, 24, 46, 90, 174, 337, 651, 1261, 2445, 4753, 9258, 18101, 35487, 69823, 137704, 272366, 539797, 1071969, 2132017, 4245964, 8464289, 16887427, 33713589, 67336900, 134542546, 268894341, 537515903, 1074640717, 2148733325, 4296686409, 8592299548, 17183084263, 34364120060, 68725368752, 137446915007, 274888501928, 549770021804, 1099530342380, 2199048203425, 4398079052052, 8796136153039, 17592241805077, 35184445671235

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
The a(0) = 0 through a(4) = 12 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}

This is the "bi-" version of A371789, differences A371790.
The complement is counted by A371791, differences A232466.
First differences are A371793.
The complement is the "bi-" version of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],Not@*biqQ]],{n,0,10}]

a(16) onwards from Martin Fuller, Mar 21 2025

A371794 Number of non-biquanimous strict integer partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 27, 23, 38, 30, 54, 43, 76, 57, 104, 79, 142, 102, 192, 138, 256, 174, 340, 232, 448, 292, 585, 375, 760, 471, 982, 602, 1260, 741, 1610, 935, 2048, 1148, 2590, 1425, 3264, 1733, 4097, 2137, 5120, 2571, 6378

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The a(1) = 1 through a(11) = 12 strict partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (521)  (81)   (91)   (92)
                                                  (432)  (631)  (A1)
                                                  (531)  (721)  (542)
                                                  (621)         (632)
                                                                (641)
                                                                (731)
                                                                (821)
                                                                (5321)

The complement is counted by A237258 aerated, ranks A357854.
Even bisection is A321142, odd A078408.
This is the "bi-" version of A371736, complement A371737.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!biqQ[#]&]],{n,0,30}]

A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 2, 1, 6, 6, 1, 6, 8, 1, 8, 18, 1, 8, 16, 1, 10, 30, 4, 1, 10, 34, 14, 1, 12, 48, 28, 1, 12, 48, 42, 1, 14, 72, 76, 1, 14, 72, 100, 1, 16, 96, 160, 8, 1, 16, 98, 190, 8, 1, 18, 126, 284, 40, 1, 18, 128, 316, 70

Offset: 1

Gus Wiseman, May 13 2019

Also the number of length-k compositions of n such that every restriction to a subinterval has a different sum. A composition of n is a finite sequence of positive integers summing to n.

			Triangle begins:
   1
   1
   1   2
   1   2
   1   4
   1   4   2
   1   6   6
   1   6   8
   1   8  18
   1   8  16
   1  10  30   4
   1  10  34  14
   1  12  48  28
   1  12  48  42
   1  14  72  76
   1  14  72 100
   1  16  96 160   8
   1  16  98 190   8
   1  18 126 284  40
   1  18 128 316  70
Row n = 8 counts the following rulers:
  {0,8}  {0,1,8}  {0,1,3,8}
         {0,2,8}  {0,1,5,8}
         {0,3,8}  {0,1,6,8}
         {0,5,8}  {0,2,3,8}
         {0,6,8}  {0,2,7,8}
         {0,7,8}  {0,3,7,8}
                  {0,5,6,8}
                  {0,5,7,8}
and the following compositions:
  (8)  (17)  (125)
       (26)  (143)
       (35)  (152)
       (53)  (215)
       (62)  (251)
       (71)  (341)
             (512)
             (521)

Eric Weisstein's World of Mathematics, Golomb Ruler.

Row sums are A169942.
Row lengths are A325678(n) = A143824(n + 1) - 1.
Column k = 2 is A052928.
Column k = 3 is A325686.
Rightmost column is A325683.
Cf. A000079, A007318, A103295, A108917, A143823, A325676, A325679, A325687.

DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,15},{k,n}],0,{2}]

A325858 Number of Golomb partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 25, 36, 47, 59, 78, 99, 122, 155, 195, 232, 295, 355, 432, 522, 641, 749, 919, 1076, 1283, 1506, 1802, 2067, 2470, 2835, 3322, 3815, 4496, 5070, 5959, 6736, 7807, 8849, 10266, 11499, 13326, 14928, 17140, 19193, 22037, 24519, 28106

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

We define a Golomb partition of n to be an integer partition of n such that every pair of distinct parts has a different difference.
Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.
The strict case is A325876.

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The A000041(9) - a(9) = 5 non-Golomb partitions of 9 are: (531), (432), (3321), (32211), (321111).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

The subset case is A143823.
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A108917, A325325, A325853, A325856, A325868.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]],{n,0,30}]

A325860 Number of subsets of {1..n} such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 8, 14, 28, 52, 104, 188, 308, 548, 1096, 1784, 3568, 6168, 10404, 16200, 32400, 49968, 99936, 155584, 256944, 433736, 867472, 1297504, 2026288, 3387216, 5692056, 8682912, 17365824, 25243200, 50486400, 78433056, 125191968, 206649216, 328195632

Offset: 0

Gus Wiseman, May 31 2019

nonn

Also subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}    {}     {}
      {1}  {1}   {1}    {1}
           {2}   {2}    {2}
           {12}  {3}    {3}
                 {12}   {4}
                 {13}   {12}
                 {23}   {13}
                 {123}  {14}
                        {23}
                        {24}
                        {34}
                        {123}
                        {134}
                        {234}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..50

The subset case is A325860.
The maximal case is A325861.
The integer partition case is A325853.
The strict integer partition case is A325854.
Heinz numbers of the counterexamples are given by A325994.
Cf. A002033, A108917, A143823, A196723, A196723, A196724, A325855, A325858, A325859, A325868, A325869.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,0,20}]

a(21)-a(25) from Alois P. Heinz, Jun 07 2019

a(26)-a(35) from Fausto A. C. Cariboni, Oct 04 2020

A325864 Number of subsets of {1..n} of which every subset has a different sum.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 56, 91, 135, 211, 307, 446, 625, 882, 1194, 1677, 2238, 3031, 4001, 5460, 6995, 9302, 11921, 15424, 19554, 25032, 31005, 39170, 48251, 59917, 73093, 90831, 109271, 134049, 160922, 196109, 234179, 284157, 335933, 408390, 482597, 575109

Offset: 0

Gus Wiseman, Jun 01 2019

nonn

			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}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..150 (terms 0..60 from Giovanni Resta)

Cf. A002033, A108917, A143823, A196723, A275972.

Cf. A325860, A325865, A325866, A325867, A325877, A325878, A325879, A325880.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#]&]],{n,0,10}]

a(18)-a(42) from Alois P. Heinz, Jun 03 2019

cp's OEIS Frontend

A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

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A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

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A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

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A371796 Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

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A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

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A371794 Number of non-biquanimous strict integer partitions of n.

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A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

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A325858 Number of Golomb partitions of n.

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