A108917 -id:A108917 - cp's OEIS frontend

A299729 Heinz numbers of non-knapsack partitions.

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer partition is non-knapsack if there exist two different submultisets with the same sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			12 is the Heinz number of (2,1,1) which is not knapsack because 2 = 1 + 1.

Cf. A056239, A108917, A112798, A275972, A276024, A284640, A296150, A299701, A299702.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@Plus@@@Union[Rest@Subsets[primeMS[#]]]&]

A292886 Number of knapsack 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, 7, 2, 2, 3, 4, 1, 5, 1, 6, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 11, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Sep 26 2017

nonn

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

The sequence giving the number of factorizations of n is described as "the multiplicative partition function" (see A001055), so knapsack factorizations are a multiplicative generalization of knapsack partitions. - Gus Wiseman, Oct 24 2017

			The a(36) = 8 factorizations are 2*2*3*3, 2*2*9, 2*18, 3*3*4, 3*12, 4*9, 6*6, 36. The factorization 2*3*6 is not knapsack.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A001055, A045778, a(p^n) = A108917(n), A162247, A259936, A275972, A281116.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[postfacs[n],UnsameQ@@Times@@@Union[Subsets[#]]&]],{n,100}]

A321142 Number of strict integer partitions of 2*n with no subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 15, 23, 30, 43, 57, 79, 102, 138, 174, 232, 292, 375, 471, 602, 741, 935, 1148, 1425, 1733, 2137, 2571, 3156, 3789, 4557, 5470, 6582, 7796, 9317, 11027, 13058, 15400, 18159, 21249, 24971, 29170, 33986, 39596, 46073, 53219, 61711, 71330, 82171

Offset: 0

Gus Wiseman, Oct 28 2018

nonn

			The a(1) = 1 through a(8) = 23 partitions:
  (2)  (4)    (6)    (8)      (10)     (12)     (14)       (16)
       (3,1)  (4,2)  (5,3)    (6,4)    (7,5)    (8,6)      (9,7)
              (5,1)  (6,2)    (7,3)    (8,4)    (9,5)      (10,6)
                     (7,1)    (8,2)    (9,3)    (10,4)     (11,5)
                     (5,2,1)  (9,1)    (10,2)   (11,3)     (12,4)
                              (6,3,1)  (11,1)   (12,2)     (13,3)
                              (7,2,1)  (5,4,3)  (13,1)     (14,2)
                                       (7,3,2)  (6,5,3)    (15,1)
                                       (7,4,1)  (8,4,2)    (7,5,4)
                                       (8,3,1)  (8,5,1)    (7,6,3)
                                       (9,2,1)  (9,3,2)    (9,4,3)
                                                (9,4,1)    (9,5,2)
                                                (10,3,1)   (9,6,1)
                                                (11,2,1)   (10,4,2)
                                                (8,3,2,1)  (10,5,1)
                                                           (11,3,2)
                                                           (11,4,1)
                                                           (12,3,1)
                                                           (13,2,1)
                                                           (6,5,4,1)
                                                           (7,4,3,2)
                                                           (9,4,2,1)
                                                           (10,3,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A002219, A006827, A108917, A237258, A275972, A279791, A284640, A305551.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!Or@@Table[SameQ[Total[#[[s]]],n/2],{s,Subsets[Range[Length[#]]]}]]&]],{n,2,20,2}]

a(33)-a(48) from Giovanni Resta, Oct 30 2018

A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 46, 58, 77, 93, 122, 146, 188, 217, 282, 327, 410, 470, 596, 673, 848, 947, 1178, 1325, 1629, 1798, 2213, 2444, 2962, 3247, 3935, 4292, 5149, 5579, 6674, 7247, 8590, 9221, 10964, 11804, 13870, 14843, 17480, 18675, 21866

Offset: 0

Gus Wiseman, May 31 2019

nonn

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

			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 three non-knapsack partitions counted under a(6) are:
  (2,2,1,1)
  (3,1,1,1)
  (2,1,1,1,1)

Dominates A108917.

Cf. A002033, A034444, A196723, A275972, A276024, A299702, A325592, A325856, A325863, A325864, A325865, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#]]&]],{n,0,20}]

A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Gus Wiseman, Jun 06 2022

nonn

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.

In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    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}
The sequence contains 18 because its prime-power divisors {1,2,3,9} have prime indices {}, {1}, {2}, {2,2} with distinct sums {0,1,2,4}. On the other hand, 12 is not in the sequence because {2} and {1,1} have the same sum.

Knapsack partitions are counted by A108917, ranked by A299702.
The strong case is A353838, counted by A353837, complement A353839.
These partitions are counted by A353864.
The complete case is A353867, counted by A353865.
The complement is A354583.
A000041 counts partitions, strict A000009.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.

msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Select[msubs[primeMS[#]],SameQ@@#&]&]

A335457 Number of normal patterns contiguously matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 31, 80, 196, 486, 1171, 2787, 6564, 15323, 35403, 81251, 185087, 418918, 942525, 2109143, 4695648, 10405694, 22959156

Offset: 0

Gus Wiseman, Jun 23 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(0) = 1 through a(3) = 12 pairs of a composition with a contiguously matched pattern:
  ()()  (1)()   (2)()     (3)()
        (1)(1)  (11)()    (12)()
                (2)(1)    (21)()
                (11)(1)   (3)(1)
                (11)(11)  (111)()
                          (12)(1)
                          (21)(1)
                          (111)(1)
                          (12)(12)
                          (21)(21)
                          (111)(11)
                          (111)(111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions is A335458.
The non-contiguous version is A335456.
Patterns are counted by A000670 and ranked by A333217.
The n-th standard composition has A124771(n) contiguous subsequences.
Patterns contiguously matched by prime indices are A335549.
Minimal avoided patterns of prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A124767, A181796, A269134, A333224, A334299.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[cmp,{_,s___,_}:>{s}]]],{cmp,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(16)-a(20) from Jinyuan Wang, Jul 08 2020

A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 6, 4, 7, 6, 5, 2, 4, 4, 6, 4, 6, 7, 8, 4, 7, 6, 10, 6, 10, 8, 6, 2, 4, 4, 6, 3, 8, 8, 8, 4, 8, 4, 9, 8, 12, 11, 10, 4, 7, 8, 10, 8, 11, 12, 13, 6, 10, 9, 14, 8, 13, 10, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 12, 7, 14, 12, 10, 4

Offset: 0

Gus Wiseman, Jun 01 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 6 4 7 6 5
  2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the subsequences of the composition with STC-number n:
  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
     0  0  1  0  2  2  3  0  4   2   5   4   6   6   7
           0     1  1  1     1   0   3   1   5   3   3
                 0  0  0     0       2   0   3   2   1
                                     1       2   1   0
                                     0       1   0
                                             0

Row lengths are A011782.
Looking only at contiguous subsequences gives A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Disallowing empty subsequences gives A334300.
Subsequence-sums are counted by A334968.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Subsets[stc[n]]]],{n,0,100}]

a(n) = A334300(n) + 1.

A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			42 is the Heinz number of (4,2,1) which is strict and knapsack, so is in the sequence. 45 is the Heinz number of (3,2,2) which is knapsack but not strict, so is not in the sequence. 30 is the Heinz number of (3,2,1) which is strict but not knapsack, so is not in the sequence.
Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301900.

wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
Select[Range[100],SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]

Intersection of A299702 and A005117.

A371795 Number of non-biquanimous integer partitions of n.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922, 6842, 1172, 10143, 1745, 14883, 2108, 21637, 3104, 31185, 3737, 44583, 5232, 63261, 6419, 89134, 8988, 124754, 10390, 173525

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(8) = 8 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)    (7)        (8)
            (21)   (31)  (32)     (42)   (43)       (53)
            (111)        (41)     (51)   (52)       (62)
                         (221)    (222)  (61)       (71)
                         (311)    (411)  (322)      (332)
                         (2111)          (331)      (521)
                         (11111)         (421)      (611)
                                         (511)      (5111)
                                         (2221)
                                         (3211)
                                         (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)

The complement is counted by A002219 aerated, ranks A357976.
Even bisection is A006827, odd A058695.
The strict complement is A237258, ranks A357854.
This is the "bi-" version of A321451, ranks A321453.
The complement is the "bi-" version of A321452, ranks A321454.
These partitions have ranks A371731.
The strict case is A371794, bisections A321142, A078408.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371736 counts non-quanimous strict partitons, complement A371737.
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. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n],Not@*biqQ]],{n,0,15}]

a(n) = if(n%2, numbpart(n), my(v=partitions(n/2), w=List([])); for(i=1, #v, for(j=1, i, listput(w, vecsort(concat(v[i], v[j]))))); numbpart(n)-#Set(w)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A325780 Heinz numbers of perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Gus Wiseman, May 21 2019

nonn

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

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}

Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).

cp's OEIS Frontend

A299729 Heinz numbers of non-knapsack partitions.

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A292886 Number of knapsack factorizations of n.

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A321142 Number of strict integer partitions of 2*n with no subset summing to n.

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A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

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A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

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A335457 Number of normal patterns contiguously matched by compositions of n.

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A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

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A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

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A371795 Number of non-biquanimous integer partitions of n.