A299702 -id:A299702 - cp's OEIS frontend

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 4, 3, 4, 4, 4, 1, 3, 3, 5, 3, 5, 4, 5, 3, 4, 5, 5, 5, 5, 5, 5, 1, 3, 3, 5, 2, 5, 5, 6, 3, 6, 3, 6, 5, 6, 5, 6, 3, 4, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 7, 4, 6, 6, 7, 3, 6, 4, 7, 5, 7, 6

Offset: 0

Gus Wiseman, Mar 18 2020

nonn

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 composition (4,3,1,2) has positive subsequence-sums 1, 2, 3, 4, 6, 7, 8, 10, so a(550) = 8.

Dominated by A124770.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702.
Strict knapsack partitions are counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Allowing empty subsequences gives A333257.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A124767, A143823, A295235, A325680, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333257(n) - 1.

A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 20 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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 ninth composition in standard order is (3,1), which has consecutive subsequences (), (1), (3), (3,1), with sums 0, 1, 3, 4, so a(9) = 4.

Dominated by A124771.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222, while the case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223.
The version for Heinz numbers of partitions is A325770.
Not allowing empty subsequences gives A333224.
Cf. A000120, A029931, A048793, A059519, A066099, A070939, A114994, A124765, A124767, A233564, A272919, A325778, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333224(n) + 1.

A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 5, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 6, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 6, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 6, 3, 7, 1, 5, 3, 6, 1, 7, 1, 3, 5, 5, 3, 7, 1, 7, 4, 3, 1, 8, 3, 3, 3, 7, 1, 8, 3, 5, 3, 3, 3, 7, 1, 5, 5, 8, 1, 7, 1, 7, 7

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive integer n is a positive subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
a(n) <= A000005(n).
One less than the number of distinct values obtained when A056239 is applied to all divisors of n. - Antti Karttunen, Jul 01 2018

			The positive subset-sums of (4,3,1) are {1, 3, 4, 5, 7, 8} so a(70) = 6.
The positive subset-sums of (5,1,1,1) are {1, 2, 3, 5, 6, 7, 8} so a(88) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A056239, A122768, A276024, A284640, A296150, A299701, A299702, A301855, A301935, A301957, A304792, A304795, A305611.

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A304793(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v056239[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jul 01 2018

More terms from Antti Karttunen, Jul 01 2018

A299729 Heinz numbers of non-knapsack partitions.

Original entry on oeis.org

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[#]]]&]

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@@#&]&]

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).

A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

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: ()            21: (2,2,1)           65: (6,1)
    1: (1)           24: (1,4)             66: (5,2)
    2: (2)           26: (1,2,2)           67: (5,1,1)
    3: (1,1)         28: (1,1,3)           68: (4,3)
    4: (3)           31: (1,1,1,1,1)       69: (4,2,1)
    5: (2,1)         32: (6)               70: (4,1,2)
    6: (1,2)         33: (5,1)             71: (4,1,1,1)
    7: (1,1,1)       34: (4,2)             72: (3,4)
    8: (4)           35: (4,1,1)           73: (3,3,1)
    9: (3,1)         36: (3,3)             74: (3,2,2)
   10: (2,2)         40: (2,4)             80: (2,5)
   12: (1,3)         41: (2,3,1)           81: (2,4,1)
   15: (1,1,1,1)     42: (2,2,2)           84: (2,2,3)
   16: (5)           48: (1,5)             85: (2,2,2,1)
   17: (4,1)         50: (1,3,2)           88: (2,1,4)
   18: (3,2)         56: (1,1,4)           96: (1,6)
   19: (3,1,1)       63: (1,1,1,1,1,1)     98: (1,4,2)
   20: (2,3)         64: (7)              100: (1,3,3)

Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{_,s__,_}:>{s}]]&]

cp's OEIS Frontend

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

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A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

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A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

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A299729 Heinz numbers of non-knapsack partitions.

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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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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.

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