A353849 -id:A353849 - cp's OEIS frontend

A353837 Number of integer partitions of n with all distinct run-sums.

1, 1, 2, 3, 4, 7, 10, 14, 17, 28, 35, 49, 62, 85, 107, 149, 174, 238, 305, 384, 476, 614, 752, 950, 1148, 1451, 1763, 2205, 2654, 3259, 3966, 4807, 5773, 7039, 8404, 10129, 12140, 14528, 17288, 20668, 24505, 29062, 34437, 40704, 48059, 56748, 66577, 78228

Offset: 0

Gus Wiseman, May 26 2022

nonn

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4). The first partition whose run-sums are not all distinct is (2,1,1).

			The a(0) = 1 through a(6) = 10 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (21111)
                                         (111111)

Max Alekseyev, Table of n, a(n) for n = 0..100

For multiplicities instead of run-sums we have A098859, ranked by A130091.
For equal run-sums we have A304442, ranked by A353833 (nonprime A353834).
These partitions are ranked by A353838, complement A353839.
The version for compositions is A353850, ranked by A353852.
The weak version (rucksack partitions) is A353864, ranked by A353866.
The weak perfect version is A353865, ranked by A353867.
A005811 counts runs in binary expansion.
A275870 counts collapsible partitions, ranked by A300273.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353840-A353846 pertain to partition run-sum trajectory.
A353849 counts distinct run-sums in standard compositions.
Cf. A000041, A008284, A047966, A071625, A073093, A116608, A175413, A181819, A333755, A353848, A353867.

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

a353837 = lambda n: sum( abs(BipartiteGraph( Matrix(len(p), len(D:=list(set.union(*map(lambda t: set(divisors(t)),p)))), lambda i,j: p[i]%D[j]==0) ).matching_polynomial()[len(D)-len(p)]) for p in Partitions(n,max_slope=-1) ) # Max Alekseyev, Sep 11 2023

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 10, 12, 13, 10, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 26, 20, 21, 18, 16, 32, 33, 34, 34, 32, 37, 38, 36, 40, 41, 32, 34, 44, 45, 42, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 42, 36, 37, 34, 32, 64, 65, 66, 66

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.

			As a triangle:
   0
   1
   2  2
   4  5  6  4
   8  9  8 10 12 13 10  8
  16 17 18 18 20 17 22 20 24 25 24 26 20 21 18 16
These are the standard composition numbers of the following compositions (transposed):
  ()  (1)  (2)  (3)    (4)      (5)
           (2)  (2,1)  (3,1)    (4,1)
                (1,2)  (4)      (3,2)
                (3)    (2,2)    (3,2)
                       (1,3)    (2,3)
                       (1,2,1)  (4,1)
                       (2,2)    (2,1,2)
                       (4)      (2,3)
                                (1,4)
                                (1,3,1)
                                (1,4)
                                (1,2,2)
                                (2,3)
                                (2,2,1)
                                (3,2)
                                (5)

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

Standard compositions are listed by A066099.
The version for partitions is A353832.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242. A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[stc[n]]],{n,0,100}]

A353850 Number of integer compositions of n with all distinct run-sums.

Original entry on oeis.org

1, 1, 2, 4, 5, 12, 24, 38, 52, 111, 218, 286, 520, 792, 1358, 2628, 4155, 5508, 9246, 13182, 23480, 45150, 54540, 94986, 146016, 213725, 301104, 478586, 851506, 1302234, 1775482, 2696942, 3746894, 6077784, 8194466, 12638334, 21763463, 28423976, 45309850, 62955524, 94345474

Offset: 0

Gus Wiseman, May 31 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(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (1111)  (41)
                                (113)
                                (122)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023

Joseph Likar, Table of n, a(n) for n = 0..120

For distinct parts instead of run-sums we have A032020.
For distinct multiplicities instead of run-sums we have A242882.
For distinct run-lengths instead of run-sums we have A329739, ptns A098859.
For runs instead of run-sums we have A351013.
For partitions we have A353837, ranked by A353838 (complement A353839).
For equal instead of distinct run-sums we have A353851, ptns A304442.
These compositions are ranked by A353852.
The weak version (rucksack compositions) is A354580, ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A175413 lists numbers whose binary expansion has all distinct runs.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353847 gives composition run-sum transformation.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

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

Terms a(21) and onwards from Joseph Likar, Aug 04 2023

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, read left-to-right. 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 terms together with their binary expansions and corresponding compositions begin:
     0:       0  ()
     1:       1  (1)
     2:      10  (2)
     3:      11  (1,1)
     4:     100  (3)
     7:     111  (1,1,1)
     8:    1000  (4)
    10:    1010  (2,2)
    11:    1011  (2,1,1)
    14:    1110  (1,1,2)
    15:    1111  (1,1,1,1)
    16:   10000  (5)
    31:   11111  (1,1,1,1,1)
    32:  100000  (6)
    36:  100100  (3,3)
    39:  100111  (3,1,1,1)
    42:  101010  (2,2,2)
    46:  101110  (2,1,1,2)
    59:  111011  (1,1,2,1,1)
    60:  111100  (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@Total/@Split[stc[#]]&]

A353849(a(n)) = 1.

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 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 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 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total/@Split[stc[n]],{n,0,30}]

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 0

Gus Wiseman, May 31 2022

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.

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 binary expansions and corresponding compositions begin:
   0:        0  ()
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   5:      101  (2,1)
   6:      110  (1,2)
   7:      111  (1,1,1)
   8:     1000  (4)
   9:     1001  (3,1)
  10:     1010  (2,2)
  12:     1100  (1,3)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  17:    10001  (4,1)
  18:    10010  (3,2)
  19:    10011  (3,1,1)
  20:    10100  (2,3)
  21:    10101  (2,2,1)
  23:    10111  (2,1,1,1)

The version for runs in binary expansion is A175413.
The version for parts instead of run-sums is A233564, counted A032020.
The version for run-lengths instead of run-sums is A351596, counted A329739.
The version for runs instead of run-sums is A351290, counted by A351013.
The version for partitions is A353838, counted A353837, complement A353839.
The equal instead of distinct version is A353848, counted by A353851.
These compositions are counted by A353850.
The weak version (rucksack compositions) is A354581, counted by A354580.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A242882 counts composition with distinct multiplicities, partitions A098859.
A304442 counts partitions with all equal run-sums.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, perfect A353865.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Split[stc[#]]&]

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

Original entry on oeis.org

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

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}]

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

A353835 Number of distinct run-sums of the prime indices of n.

Original entry on oeis.org

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

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.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 prime indices of 3780 are {1,1,2,2,2,3,4}, with distinct run-sums {2,3,4,6}, so a(3780) = 4.
The prime indices of 8820 are {1,1,2,2,3,4,4}, with distinct run-sums {2,3,4,8}, so a(8820) = 4.
The prime indices of 13860 are {1,1,2,2,3,4,5}, with distinct run-sums {2,3,4,5}, so a(13860) = 4.
The prime indices of 92400 are {1,1,1,1,2,3,3,4,5}, with distinct run-sums {2,4,5,6}, so a(92400) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A002110.
A version for binary expansion is A165413.
Positions of 0's and 1's are A353833, nonprime A353834, counted by A304442.
The case of all distinct run-sums is ranked by A353838, counted by A353837.
The version for compositions is A353849.
The weak version is A353861.
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.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353839, A353848, A353850, A353852, A353867.

Table[Length[Union[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k]]],{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); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
A353835(n) = omega(A353832(n)); \\ Antti Karttunen, Jan 20 2025

a(n) = A001221(A353832(n)). [From formula section of A353832] - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A353837 Number of integer partitions of n with all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A353850 Number of integer compositions of n with all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments