A325857 -id:A325857 - cp's OEIS frontend

A196723 Number of subsets of {1..n} (including empty set) such that the pairwise sums of distinct elements are all distinct.

1, 2, 4, 8, 15, 28, 50, 86, 143, 236, 376, 594, 913, 1380, 2048, 3016, 4367, 6302, 8974, 12670, 17685, 24580, 33738, 46072, 62367, 83990, 112342, 149734, 198153, 261562, 343210, 448694, 583445, 756846, 976086, 1255658, 1607831, 2053186, 2610560, 3312040, 4183689

Offset: 0

Alois P. Heinz, Oct 06 2011

nonn

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum is A143823(n).

			a(4) = 15: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..110

Cf. A143823, A196719, A196720, A196721, A196722, A196724.
The subset case is A196723 (this sequence).
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A108917, A325858, A325862, A325863, A325864.

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       sn[i]+sn[j], j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n<1, 1, b[n-1, s] + If[m*(m+1)/2 == Length[ Union[ Flatten[ Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i+1, m+1}]]]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Edited by Gus Wiseman, Jun 03 2019

A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 18, 19, 26, 28, 36, 37, 50, 52, 67, 68, 89, 94, 115, 121, 151, 160, 195, 200, 247, 265, 312, 329, 386, 418, 487, 519, 600, 640, 742, 792, 901, 978, 1088, 1185, 1331, 1453, 1605, 1729, 1925, 2101, 2311, 2524, 2741, 3000

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

The non-strict case is A325857.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)   (43)   (53)   (54)   (64)
                        (41)  (51)   (52)   (62)   (63)   (73)
                              (321)  (61)   (71)   (72)   (82)
                                     (421)  (431)  (81)   (91)
                                            (521)  (432)  (532)
                                                   (531)  (541)
                                                   (621)  (631)
                                                          (721)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A275972.
Cf. A325854, A325855, A325858, A325862, A325864, A325876, A325879.

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

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 8, 22, 40, 56, 78, 124, 222, 390, 616, 892, 1220, 1620, 2182, 3042, 4392, 6364, 9054, 12608, 16980, 22244, 28482, 36208, 45864, 58692, 75804, 98440, 128694, 168250, 218558, 281210, 357594, 449402, 560034, 693332, 853546, 1050118, 1293458, 1596144, 1975394

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..60

The subset case is A196723.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A143824, A276024.
Cf. A325858, A325859, A325864, A325865, A325867, A325879, A325880, A382398.

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

a(n)={
   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,b< n, ismaxl(b,w),
         my(s=self()(k+1, r, b, w));
         if(!bitand(w,b<Andrew Howroyd, Mar 23 2025

a(21) onwards from Andrew Howroyd, Mar 23 2025

A325856 Number of integer partitions of n such that every pair of distinct parts has a different product.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 171, 225, 287, 369, 467, 592, 740, 931, 1155, 1435, 1767, 2178, 2661, 3254, 3953, 4798, 5793, 6991, 8390, 10069, 12022, 14346, 17054, 20255, 23960, 28334, 33390, 39308, 46148, 54116, 63295, 73967, 86224

Offset: 0

Gus Wiseman, May 31 2019

nonn

			The five partitions of 15 not satisfying the condition are:
  (8,4,2,1)
  (6,4,3,2)
  (6,3,3,2,1)
  (6,3,2,2,1,1)
  (6,3,2,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..150

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Cf. A002033, A108917, A143823, A325853, A325854, A325857, A325858, A325877.

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

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 1, 1, 0, 1, 1, 2, 2, 4, 3, 2, 1, 1, 0, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 0, 1, 1, 3, 3, 4, 6, 4, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 6, 4

Offset: 0

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

210, 420, 462, 630, 840, 858, 910, 924, 1050, 1155, 1260, 1326, 1386, 1470, 1680, 1716, 1820, 1848, 1870, 1890, 1938, 2100, 2145, 2310, 2470, 2520, 2574, 2622, 2652, 2730, 2772, 2926, 2940, 3150, 3234, 3315, 3360, 3432, 3465, 3570, 3640, 3696, 3740, 3780, 3876

Offset: 1

Gus Wiseman, Jun 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   210: {1,2,3,4}
   420: {1,1,2,3,4}
   462: {1,2,4,5}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}
   858: {1,2,5,6}
   910: {1,3,4,6}
   924: {1,1,2,4,5}
  1050: {1,2,3,3,4}
  1155: {2,3,4,5}
  1260: {1,1,2,2,3,4}
  1326: {1,2,6,7}
  1386: {1,2,2,4,5}
  1470: {1,2,3,4,4}
  1680: {1,1,1,1,2,3,4}
  1716: {1,1,2,5,6}
  1820: {1,1,3,4,6}
  1848: {1,1,1,2,4,5}
  1870: {1,3,5,7}
  1890: {1,2,2,2,3,4}

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A056239, A103300, A108917, A112798, A143823, A196724, A325853, A325855, A325858, A325859, A325862, A325992, A325993, A325994.

Select[Range[1000],!UnsameQ@@Plus@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

1925, 12155, 20995, 23375, 37145

Offset: 1

Gus Wiseman, Jun 03 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition is knapsack if every submultiset has a different sum.
The enumeration of these partitions by sum is given by A326016.

			The sequence of terms together with their prime indices begins:
   1925: {3,3,4,5}
  12155: {3,5,6,7}
  20995: {3,6,7,8}
  23375: {3,3,3,5,7}
  37145: {3,7,8,9}

Cf. A002033, A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325782, A325857, A325862, A325878, A325880, A326015, A326016.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]];
Select[Range[2,200],With[{phm=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},ksQ[phm]&&Select[Table[Sort[Append[phm,i]],{i,Max@@phm}],ksQ]=={}]&]

A364465 Number of subsets of {1..n} with all different first differences of elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 61, 99, 156, 240, 381, 587, 894, 1334, 1967, 2951, 4370, 6406, 9293, 13357, 18976, 27346, 39013, 55437, 78154, 109632, 152415, 210801, 293502, 406664, 561693, 772463, 1058108, 1441796, 1956293, 2639215, 3579542, 4835842, 6523207

Offset: 0

Gus Wiseman, Jul 30 2023

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}
                         {1,3,4}

Rémy Sigrist, C++ program

For all differences of pairs of elements we have A196723
For partitions instead of subsets we have A325325, strict A320347.
For subset-sums we have A325864, for partitions A108917, A275972.
A007318 counts subsets by length.
A053632 counts subsets by sum.
A363260 counts partitions disjoint from differences, complement A364467.
A364463 counts subsets disjoint from differences, complement A364466.
Cf. A000009, A008289, A011782, A236912, A320348, A325857, A325877, A325878, A326083, A364345, A364346.

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

More terms from Rémy Sigrist, Aug 06 2023

A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 4, 5, 0, 30, 2, 9, 9, 20, 3, 37, 6, 18, 16, 37, 20, 71, 12, 37, 40

Offset: 1

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Union[#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

cp's OEIS Frontend

A196723 Number of subsets of {1..n} (including empty set) such that the pairwise sums of distinct elements are all distinct.

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A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

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A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

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A325856 Number of integer partitions of n such that every pair of distinct parts has a different product.

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A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

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A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

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A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

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A364465 Number of subsets of {1..n} with all different first differences of elements.

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