A353832 -id:A353832 - cp's OEIS frontend

A353851 Number of integer compositions of n with all equal run-sums.

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 2022

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(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}]

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 5, 9, 1, 0, 0, 0, 0, 0, 6, 11, 4, 1, 0, 0, 0, 0, 0, 8, 20, 2, 0, 0, 0, 0, 0, 0, 0, 10, 25, 7, 0, 0, 0, 0, 0, 0, 0, 0, 12, 37, 6, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, May 26 2022

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 run-sum trajectory is obtained by repeatedly taking run-sums (or condensations) until a strict partition is reached. For example, the trajectory of (2,1,1) is (2,1,1) -> (2,2) -> (4).

Also the number of integer partitions of n with Kimberling's depth statistic (see A237685, A237750) equal to k-1.

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   1   0
   0   2   2   1   0
   0   3   4   0   0   0
   0   4   6   1   0   0   0
   0   5   9   1   0   0   0   0
   0   6  11   4   1   0   0   0   0
   0   8  20   2   0   0   0   0   0   0
   0  10  25   7   0   0   0   0   0   0   0
   0  12  37   6   1   0   0   0   0   0   0   0
   0  15  47  13   2   0   0   0   0   0   0   0   0
   0  18  67  15   1   0   0   0   0   0   0   0   0   0
   0  22  85  25   3   0   0   0   0   0   0   0   0   0   0
   0  27 122  26   1   0   0   0   0   0   0   0   0   0   0   0
For example, row n = 8 counts the following partitions (empty columns indicated by dots):
.  (8)    (44)        (422)     (4211)  .  .  .  .
   (53)   (332)       (32111)
   (62)   (611)       (41111)
   (71)   (2222)      (221111)
   (431)  (3221)
   (521)  (3311)
          (5111)
          (22211)
          (311111)
          (2111111)
          (11111111)

Row-sums are A000041.
Column k = 1 is A000009.
Column k = 2 is A237685.
Column k = 3 is A237750.
The version for run-lengths instead of run-sums is A225485 or A325280.
This statistic (trajectory length) is ranked by A353841 and A326371.
The version for compositions is A353859, see also A353847-A353858.
A005811 counts runs in binary expansion.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition
A353836 counts partitions by number of distinct run-sums.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353845 counts partitions whose run-sum trajectory ends in a singleton.
Cf. A008284, A047966, A181819, A325239, A325277, A353834, A353865.

rsn[y_]:=If[y=={},{},NestWhileList[Reverse[Sort[Total/@ Split[Sort[#]]]]&,y,!UnsameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],Length[rsn[#]]==k&]],{n,0,15},{k,0,n}]

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 3, 3, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3

Offset: 0

Gus Wiseman, May 30 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 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.

			Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.

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

Counting repeated runs also gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of run-sums we have A351014 (firsts A351015).
A version for partitions is A353835, weak A353861.
Positions of 1's are A353848, counted by A353851.
The version for binary expansion is A353929 (firsts A353930).
The run-sums themselves are listed by A353932, with A353849 distinct terms.
For distinct run-lengths instead of run-sums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct run-lengths in binary expansion.
A297770 counts distinct runs in binary expansion, firsts A350952.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

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

A353853 Trajectory of the composition run-sum transformation (or condensation) of n, using standard composition numbers.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 5, 6, 7, 4, 8, 9, 10, 8, 11, 10, 8, 12, 13, 14, 10, 8, 15, 8, 16, 17, 18, 19, 18, 20, 21, 17, 22, 23, 20, 24, 25, 26, 24, 27, 26, 24, 28, 20, 29, 21, 17, 30, 18, 31, 16, 32, 33, 34, 35, 34, 36, 32, 37, 38, 39, 36, 32, 40, 41, 42, 32

Offset: 0

Gus Wiseman, Jun 01 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 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.
The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 given in row 11 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).

			Triangle begins:
   0
   1
   2
   3  2
   4
   5
   6
   7  4
   8
   9
  10  8
  11 10  8
  12
  13
  14 10  8
For example, the trajectory of 29 is 29 -> 21 -> 17, corresponding to the compositions (1,1,2,1) -> (2,2,1) -> (4,1).

These sequences for partitions are A353840-A353846.
This is the iteration of A353847, with partition version A353832.
Row-lengths are A353854, counted by A353859.
Final terms are A353855.
Counting rows by weight of final term gives A353856.
Rows ending in a power of 2 are A353857, counted by A353858.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A329739 counts compositions with all distinct run-lengths.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353929 counts distinct runs in binary expansion, firsts A353930.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353833, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[NestWhileList[stcinv[Total/@Split[stc[#]]]&,n,MatchQ[stc[#],{_,x_,x_,_}]&],{n,0,50}]

A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

Offset: 1

Gus Wiseman, May 25 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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.

			Triangle begins:
   1
   2
   3
   4  3
   5
   6
   7
   8  5
   9  7
  10
  11
  12  9  7
Row 87780 is the following trajectory (left column), with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}

The version for run-lengths instead of sums is A325239 or A325277.
This is the iteration of A353832, with composition version A353847.
Row-lengths are A353841, counted by A353846.
Final terms are A353842.
Counting rows by final omega gives A353843.
Rows ending in a prime number are A353844, counted by A353845.
These sequences for compositions are A353853-A353859.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A182850 or A323014 gives frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353862 gives greatest run-sum of prime indices, least A353931.
Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]

A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 4, 2, 2, 0, 0, 7, 7, 2, 0, 0, 0, 14, 14, 4, 0, 0, 0, 0, 23, 29, 12, 0, 0, 0, 0, 0, 39, 56, 25, 8, 0, 0, 0, 0, 0, 71, 122, 53, 10, 0, 0, 0, 0, 0, 0, 124, 246, 126, 16, 0, 0, 0, 0, 0, 0, 0, 214, 498, 264, 48, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 02 2022

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 run-sum trajectory is obtained by repeatedly taking the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,4,2,1,1) -> (2,4,2,2) -> (2,4,4) -> (2,8) is counted under T(10,4).

			Triangle begins:
   1
   0   1
   0   1   1
   0   3   1   0
   0   4   2   2   0
   0   7   7   2   0   0
   0  14  14   4   0   0   0
   0  23  29  12   0   0   0   0
   0  39  56  25   8   0   0   0   0
   0  71 122  53  10   0   0   0   0   0
   0 124 246 126  16   0   0   0   0   0   0
   0 214 498 264  48   0   0   0   0   0   0   0
For example, row n = 5 counts the following compositions:
  (5)    (113)    (1121)
  (14)   (122)    (1211)
  (23)   (221)
  (32)   (311)
  (41)   (1112)
  (131)  (2111)
  (212)  (11111)

Column k = 1 is A003242, ranked by A333489, complement A261983.
Row sums are A011782.
Positive row-lengths are A070939.
The version for partitions is A353846, ranked by A353841.
This statistic (trajectory length) is ranked by A353854, firsts A072639.
Counting by length of last part instead of number of parts gives A353856.
A333627 ranks the run-lengths of standard compositions.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.

rsc[y_]:=If[y=={},{},NestWhileList[Total/@Split[#]&,y,MatchQ[#,{_,x_,x_,_}]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[rsc[#]]==k&]],{n,0,10},{k,0,n}]

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 4, 3, 4, 1, 3, 5, 2, 2, 6, 1, 4, 2, 3, 4, 7, 1, 4, 8, 2, 3, 2, 4, 1, 5, 9, 3, 2, 6, 1, 6, 6, 2, 4, 10, 1, 2, 3, 11, 5, 2, 5, 1, 7, 3, 4, 2, 4, 12, 1, 8, 2, 6, 3, 3, 13, 1, 2, 4, 14, 2, 5, 4, 3, 1, 9, 15, 4, 2, 8, 1, 6, 2, 7, 2, 6, 16

Offset: 1

Gus Wiseman, Jun 17 2022

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.

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).

			Triangle begins:
  .
  1
  2
  2
  3
  1 2
  4
  3
  4
  1 3
  5
  2 2
  6
  1 4
  2 3
For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).

Positions of first appearances are A308495 plus 1.
The version for compositions is A353932, ranked by A353847.
Classes:
- singleton rows: A000961
- constant rows: A353833, nonprime A353834, counted by A304442
- strict rows: A353838, counted by A353837, complement A353839
Statistics:
- row lengths: A001221
- row sums: A056239
- row products: A304117
- row ranks (as partitions): A353832
- row image sizes: A353835
- row maxima: A353862
- row minima: A353931
A001222 counts prime factors with multiplicity.
A112798 and A296150 list partitions by rank.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct sums of partial runs of prime indices.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.

Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,30}]

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@@#&]&]

A353839 Numbers whose prime indices do not have all distinct run-sums.

Original entry on oeis.org

12, 40, 60, 63, 84, 112, 120, 126, 132, 144, 156, 204, 228, 252, 276, 280, 300, 315, 325, 336, 348, 351, 352, 360, 372, 420, 440, 444, 492, 504, 516, 520, 560, 564, 588, 630, 636, 650, 660, 675, 680, 693, 702, 708, 720, 732, 760, 780, 804, 819, 832, 840, 852

Offset: 1

Gus Wiseman, Jun 04 2022

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.

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 terms together with their prime indices begin:
   12: {1,1,2}
   40: {1,1,1,3}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  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}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  280: {1,1,1,3,4}
  300: {1,1,2,3,3}
  315: {2,2,3,4}

For equal run-sums we have A353833, counted by A304442, nonprime A353834.
The complement is A353838, counted by A353837.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353862 gives the greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002110, A071625, A073093, A116608, A118914, A124010, A353861, A353867.

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

A353861 Number of distinct weak run-sums of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 23 2022

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.

A weak run-sum of a sequence is the sum of any consecutive constant subsequence.

			The prime indices of 72 are {1,1,1,2,2}, with weak runs {}, {1}, {1,1}, {1,1,1}, {2}, {2,2}, which have sums 0, 1, 2, 3, 2, 4, of which 5 are distinct, so a(72) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of 2's are A000040.
Positions of first appearances are A000079.
The strong version is A353835, firsts A002110.
Partitions with distinct run-sums are ranked by A353838, counted by A353837.
The strong version for compositions is A353849.
The greatest run-sum is given by A353862, least A353931.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353834, A353839, A353866, A353867, A353930.

Table[Length[Union@@Cases[FactorInteger[n],{p_,k_}:>Range[0,k]*PrimePi[p]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353861(n) = if(1==n,n,my(pruns = pis_to_runs(n), runsum = 0, runsums = List([])); for(i=1,#pruns, listput(runsums, runsum); if((i>1) && pruns[i] == pruns[i-1], runsum += pruns[i], runsum = pruns[i])); listput(runsums, runsum); #Set(runsums)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A353851 Number of integer compositions of n with all equal run-sums.

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A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

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A353849 Number of distinct positive run-sums of the n-th composition in standard order.

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A353853 Trajectory of the composition run-sum transformation (or condensation) of n, using standard composition numbers.

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A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

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A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

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A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

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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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A353839 Numbers whose prime indices do not have all distinct run-sums.

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