A299702 -id:A299702 - cp's OEIS frontend

A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

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, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, Jul 26 2023

nonn

Or numbers without any prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A364345.

			We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}

Subsets of this type are counted by A007865 (sum-free sets).
Partitions of this type are counted by A364345.
The squarefree case is counted by A364346.
The complement is A364348, counted by A363225.
The non-binary version is counted by A364350.
Without re-using parts we have A364461, counted by A236912.
Without re-using parts we have complement A364462, counted by A237113.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
A323092 counts double-free partitions, ranks A320340.
Cf. A093971, A237667, A288728, A325862, A326083, A363226, A364531.

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

A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 58, 59, 61, 62, 63, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 113

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

Also numbers m whose prime indices do not have a submultiset summing to bigomega(m).
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.
These are the Heinz numbers of the partitions counted by A367213.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        29: {10}       58: {1,10}
     5: {3}        31: {11}       59: {17}
     7: {4}        34: {1,7}      61: {18}
    10: {1,3}      35: {3,4}      62: {1,11}
    11: {5}        37: {12}       63: {2,2,4}
    13: {6}        38: {1,8}      65: {3,6}
    14: {1,4}      41: {13}       67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    22: {1,5}      46: {1,9}      73: {21}
    23: {9}        47: {15}       74: {1,12}
    25: {3,3}      49: {4,4}      76: {1,1,8}
    26: {1,6}      52: {1,1,6}    77: {4,5}
    27: {2,2,2}    53: {16}       79: {22}
    28: {1,1,4}    55: {3,5}      82: {1,13}

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
A000700 counts self-conjugate partitions, ranks A088902.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A229816 counts partitions whose length is not a part, ranks A367107.
A237667 counts sum-free partitions, ranks A364531.
A365924 counts incomplete partitions, ranks A365830.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A288728, A304792, A325761, A325781, A364345, A364347.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], FreeQ[Total/@prix/@Divisors[#], PrimeOmega[#]]&]

A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 18, 38, 82, 175, 373, 787, 1651, 3439, 7126, 14667, 30049, 61249, 124440, 251922, 508779, 1025183, 2062287, 4142644, 8312927, 16667005, 33395275, 66880828, 133892910, 267976571, 536225921, 1072842931, 2146233971, 4293248183, 8587569636, 17176654105, 34355356676, 68713584720, 137430991937, 274867311960, 549741605972, 1099492913172, 2198998307679, 4398013970156, 8796049891377, 17592130283755, 35184298506429

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.

			For S = {1,3,4,6} we have {{1,6},{3,4}}, so S is counted under a(6).
The a(0) = 1 through a(6) = 18 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}
                                             {1,2,3,6}
                                             {1,2,4,5}
                                             {1,2,5,6}
                                             {1,3,4,6}
                                             {2,3,4,5}
                                             {2,3,5,6}
                                             {3,4,5,6}
                                             {1,2,3,4,6}
                                             {1,2,4,5,6}
                                             {2,3,4,5,6}

First differences are A232466.
The complement is counted by A371792, differences A371793.
This is the "bi-" case 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, A357879, A365661, A366320, A365381, A365925, A367094, A371788, A371789.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],biqQ]],{n,0,15}]

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

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

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.

These are the Heinz numbers of the partitions counted by A367218.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

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
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict partitions, counted by A000009.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A066208 ranks partitions into odd parts, counted by A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A237668 counts sum-full partitions, ranks A364532.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]

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}]

A347460 Number of distinct possible alternating products of factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 5, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
  1  4  8    12   24   30    36   48    60    120
     1  2    3    6    10/3  9    12    15    30
        1/2  3/4  8/3  5/6   4    16/3  20/3  40/3
             1/3  2/3  3/10  1    3     15/4  15/2
                  3/8  2/15  4/9  3/4   12/5  24/5
                  1/6        1/4  1/3   3/5   10/3
                             1/9  3/16  5/12  5/6
                                  1/12  4/15  8/15
                                        3/20  3/10
                                        1/15  5/24
                                              2/15
                                              3/40
                                              1/30

Positions of 1's are 1 and A000040.
Positions of 2's appear to be A001358.
Positions of 3's appear to be A030078.
Dominates A038548, the version for reverse-alternating product.
Counting only integers gives A046951.
The even-length case is A072670.
The version for partitions (not factorizations) is A347461, reverse A347462.
The odd-length case is A347708.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A292886 counts knapsack factorizations, by sum A293627.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
Cf. A002033, A119620, A143823, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@facs[n]]],{n,100}]

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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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