A353833 -id:A353833 - cp's OEIS frontend

A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Gus Wiseman, May 26 2022

nonn

The run-sums transformation is described by Kimberling at A237685 and A237750.
The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.

			The terms together with their prime indices begin:
      1: {}            25: {3,3}           64: {1,1,1,1,1,1}
      2: {1}           27: {2,2,2}         67: {19}
      3: {2}           29: {10}            71: {20}
      4: {1,1}         31: {11}            73: {21}
      5: {3}           32: {1,1,1,1,1}     79: {22}
      7: {4}           37: {12}            81: {2,2,2,2}
      8: {1,1,1}       40: {1,1,1,3}       83: {23}
      9: {2,2}         41: {13}            84: {1,1,2,4}
     11: {5}           43: {14}            89: {24}
     12: {1,1,2}       47: {15}            97: {25}
     13: {6}           49: {4,4}          101: {26}
     16: {1,1,1,1}     53: {16}           103: {27}
     17: {7}           59: {17}           107: {28}
     19: {8}           61: {18}           109: {29}
     23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.

This sequence is a subset of A300273, counted by A275870.
The version for compositions is A353857, counted by A353847.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353853-A353859 pertain to composition run-sum trajectory.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.

ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]

A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 2, 8, 3, 5, 2, 15, 2, 5, 4, 18, 2, 13, 2, 14, 4, 5, 2, 62, 3, 5, 5, 14, 2, 18, 2, 48, 4, 5, 4, 71, 2, 5, 4, 54, 2, 18, 2, 14, 10, 5, 2, 374, 3, 9, 4, 14, 2, 37, 4, 54, 4, 5, 2, 131

Offset: 0

Gus Wiseman, May 26 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (3111)               (2222)
                                     (111111)             (4211)
                                                          (41111)
                                                          (221111)
                                                          (11111111)
For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Dominated by A018818 (partitions into divisors).
The version for compositions is A353858.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.

Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&,#,!UnsameQ@@#&]]<=1&]],{n,0,30}]

A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19

Offset: 0

Gus Wiseman, Jun 04 2022

The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).

			Triangle begins:
   1
   1
   2
   2  1
   4  1
   2  5
   5  5  1
   2 12  1
   8 11  3
   3 19  8
   5 27  9  1
   2 34 19  1
  15 26 34  2
   2 49 45  5
   5 68 48 14
   4 58 98 15  1
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (4211)      (611)
  (41111)     (3311)
  (221111)    (5111)
  (11111111)  (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Row-lengths are A003056.
The last part of the same trajectory is A353842.
Column k = 1 is A353845, compositions A353858.
The length of the trajectory is A353846.
The version for compositions is A353856.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
A325268 counts partitions by omicron, rank statistic A304465.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
A353932 lists run-sums of standard compositions.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
Cf. A047966, A237685, A325277, A353841, A353853-A353859.

Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]

A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

12, 24, 36, 40, 48, 60, 63, 72, 80, 84, 96, 108, 112, 120, 126, 132, 144, 156, 160, 168, 180, 189, 192, 200, 204, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 320, 324, 325, 336, 348, 351, 352, 360, 372, 378, 384, 396, 400, 408, 420, 432, 440

Offset: 1

Gus Wiseman, Jun 15 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.

The term rucksack is short for run-knapsack.

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.

Knapsack partitions are counted by A108917, ranked by A299702.
Non-knapsack partitions are ranked by A299729.
The non-partial version is A353839, complement A353838 (counted by A353837).
The complement is A353866, counted by A353864.
The complete complement is A353867, counted by A353865.
The complement for compositions is counted by A354580.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A333223 ranks knapsack compositions, counted by A325676.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353861 counts distinct partial run-sums of prime indices.
A354584 lists run-sums of prime indices, rows ranked by A353832.
Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, A353931.

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

A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 2, 10, 3, 6, 2, 24, 2, 6, 4, 17, 2, 36, 2, 18, 4, 6, 2, 86, 3, 6, 10, 18, 2, 44, 2, 50, 4, 6, 4, 159, 2, 6, 4, 62, 2, 44, 2, 18, 30, 6, 2, 486, 3, 12, 4, 18, 2, 140, 4, 62, 4, 6, 2, 932, 2, 6, 30, 157, 4, 44, 2, 18, 4, 20, 2, 1500, 2, 6

Offset: 0

Gus Wiseman, Apr 03 2025

nonn

These are strict twice-partitions of weight n and type PRR.

			The a(1) = 1 through a(8) = 10 twice-partitions:
  (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
       (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
                    (1111)            (222)                (2222)
                    (11)(2)           (111111)             (22)(4)
                    (2)(11)           (111)(3)             (4)(22)
                                      (3)(111)             (1111)(4)
                                                           (4)(1111)
                                                           (11111111)
                                                           (1111)(22)
                                                           (22)(1111)

Gus Wiseman, Sequences enumerating triangles of integer partitions

For distinct instead of equal block-sums we have A279786.
This is the strict case of A279789.
The orderless version is A304442, see A353833, A381995, A381871.
Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.
Partitions with no partition of this type are counted by A382076, strict case of A381993.
Normal multiset partitions of this type are counted by the strict case of A382204.
A006171 counts multiset partitions into constant blocks of integer partitions of n.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.

Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]

a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Gus Wiseman, Apr 17 2025

nonn

First differs from A381871 in having 36.

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, sum A056239.

			The prime indices of 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   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}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}

For run-lengths instead of sums we have A059404, distinct A130092.
The complement is A353833, counted by A304442.
For distinct instead of equal run-sums we have A353839.
Partitions of this type are counted by A382076.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with a common run-sum, ranks A353848.
A353862 gives the greatest run-sum of prime indices, least A353931.
A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

Select[Range[100], !SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

cp's OEIS Frontend

A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

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A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

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A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

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A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

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A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

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Formula

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

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