A353837 -id:A353837 - cp's OEIS frontend

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
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)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

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

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

More terms from Bert Dobbelaere, Apr 26 2025

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173

Offset: 1

Gus Wiseman, Apr 20 2025

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, sum A056239.

			The prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Equals A383015 \/ A383099, counted by A353851 \/ A383095.

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
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)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

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

More terms from Bert Dobbelaere, Apr 26 2025

A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 11, 18, 25, 30, 41, 55, 63, 87, 98, 125, 147, 192, 213, 271, 313, 389, 444, 551, 621, 767, 874, 1055, 1209, 1444, 1646, 1965, 2244, 2644, 2991

Offset: 0

Gus Wiseman, May 18 2025

A partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

A partition is section-sum iff its conjugate is Look-and-Say, meaning it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The a(4) = 1 through a(11) = 9 partitions:
  211  221   21111  2221    422      22221     442        222221
       2111         22111   22211    222111    4222       322211
                    211111  221111   2211111   222211     332111
                            2111111  21111111  322111     422111
                                               2221111    2222111
                                               22111111   3221111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
Conjugates of the a(4) = 1 through a(11) = 9 partitions:
  (3,1)  (3,2)  (5,1)  (4,3)  (5,3)      (5,4)  (6,4)      (6,5)
         (4,1)         (5,2)  (6,2)      (6,3)  (7,3)      (7,4)
                       (6,1)  (7,1)      (7,2)  (8,2)      (8,3)
                              (3,3,1,1)  (8,1)  (9,1)      (9,2)
                                                (6,3,1)    (10,1)
                                                (3,3,2,2)  (6,3,2)
                                                (4,4,1,1)  (6,4,1)
                                                           (7,3,1)
                                                           (6,3,1,1)

Ranking sequences are shown in parentheses below.
These partitions are ranked by (A383516).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A351592 counts non Wilf Look-and-Say partitions (A384006).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383519 counts section-sum Wilf partitions (A383520).
Cf. A047966, A048767, A320348, A325324, A325325, A353837, A381431, A383511, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]=={}&]], {n,0,30}]

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

Original entry on oeis.org

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
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.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 39, 68, 125, 227, 402, 710, 1280, 2281, 4040, 7196, 12780, 22623, 40136, 71121, 125863, 222616, 393305, 695059, 1227990, 2167059, 3823029, 6743268, 11889431, 20955548, 36920415, 65030404, 114519168, 201612634, 354849227

Offset: 0

Gus Wiseman, Jun 13 2022

nonn

We define a partial run of a sequence to be any contiguous constant subsequence. The term rucksack is short for run-knapsack.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)    (3)      (4)        (5)
           (1,1)  (1,2)    (1,3)      (1,4)
                  (2,1)    (2,2)      (2,3)
                  (1,1,1)  (3,1)      (3,2)
                           (1,2,1)    (4,1)
                           (1,1,1,1)  (1,1,3)
                                      (1,2,2)
                                      (1,3,1)
                                      (2,1,2)
                                      (2,2,1)
                                      (3,1,1)
                                      (1,1,1,1,1)

Max Alekseyev, Table of n, a(n) for n = 0..65

The knapsack version is A325676, ranked by A333223.
The non-partial version for partitions is A353837, ranked by A353838 (complement A353839).
The non-partial version is A353850, ranked by A353852.
The version for partitions is A353864, ranked by A353866.
The complete version for partitions is A353865, ranked by A353867.
These compositions are ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A108917 counts knapsack partitions, ranked by A299702, strict A275972.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A353836 counts partitions by number of distinct run-sums.
A353847 is the composition run-sum transformation.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions, ranked by A354908.
Cf. A143823, A169942, A242882, A325545, A325680, A325682, A325685, A325687, A329739, A351017, A353849.

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

Terms a(16) onward from Max Alekseyev, Sep 10 2023

A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 21, 27, 30, 33, 41, 50, 57, 68, 79, 89, 112, 126, 144, 172, 198, 220, 257, 298, 327, 383, 423, 477, 533, 621, 650, 760, 816, 920, 1013

Offset: 0

Gus Wiseman, May 19 2025

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).
For non Wilf instead of Wilf we have A383506 (A383514).
These partitions are ranked by (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A354578 Number of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent divisors are equal.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 0, 0, 2, 2, 3, 0, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 4, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 0, 1, 1, 0, 0, 1, 2, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 0, 5, 2, 2, 0, 5, 1, 3, 0, 1, 1, 0, 0, 3, 3, 5, 0, 3, 1, 1

Offset: 0

Gus Wiseman, Jun 11 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). Then a(n) is the number of integer compositions whose run-sums constitute the n-th composition in standard order (graded reverse-lexicographic, A066099).

			The terms 2^(n - 1) through 2^n - 1 sum to 2^n. As a triangle:
  1
  1
  2 0
  2 1 1 0
  3 1 2 0 1 1 0 0
  2 2 3 0 3 1 1 0 2 1 1 0 0 0 0 0
The a(n) compositions for selected n:
  n=1: n=2:   n=8:       n=32:          n=68:        n=130:
----------------------------------------------------------------------
  (1)  (2)    (4)        (6)            (4,3)        (6,2)
       (1,1)  (2,2)      (3,3)          (2,2,3)      (3,3,2)
              (1,1,1,1)  (2,2,2)        (4,1,1,1)    (6,1,1)
                         (1,1,1,1,1,1)  (1,1,1,1,3)  (3,3,1,1)
                                        (2,2,1,1,1)  (2,2,2,1,1)
                                                     (1,1,1,1,1,1,2)

First column is 1 followed by A000005.
Row-sums are A011782.
Standard compositions are listed by A066099.
Positions of 0's are A354904.
Positions of first appearances are A354905.
A003242 counts anti-run compositions, ranked by A333489.
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.
A354584 gives run-sums of prime indices, rows ranked by A353832.
Cf. A029837, A124767, A175413, A238279/A333755, A333381, A353847, A353849.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,30}]

A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

Original entry on oeis.org

3, 0, 2, 8, 32, 68, 130, 290, 274, 580, 520, 1298, 2080, 1096, 2082, 4168, 2178, 4164, 4386, 35137, 8328, 8786, 10274, 8772, 16712, 20562, 8712, 16658, 33320, 41554, 33288, 82210, 34856, 66628, 33312, 66642, 34850, 69704, 140306, 133448, 69714, 74308, 133154

Offset: 0

Gus Wiseman, Jun 21 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 and their corresponding compositions begin:
     3: (1,1)
     0: ()
     2: (2)
     8: (4)
    32: (6)
    68: (4,3)
   130: (6,2)
   290: (3,4,2)
   274: (4,3,2)
   580: (3,4,3)
   520: (6,4)
  1298: (2,4,3,2)
The inverse run-sum compositions for n = 2, 8, 32, 68, 130, 290:
  (2)   (4)     (6)       (43)     (62)       (342)
  (11)  (22)    (33)      (223)    (332)      (3411)
        (1111)  (222)     (4111)   (611)      (11142)
                (111111)  (11113)  (3311)     (32211)
                          (22111)  (22211)    (111411)
                                   (1111112)  (311112)
                                              (1112211)

The standard compositions used here are A066099, run-sums A353847/A353932.
This is the position of the first appearance of n in A354578.
A011782 counts compositions.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A354904 lists positions of zeros in A354578, complement A354912.
A354908 ranks collapsible compositions, counted by A353860.
Cf. A000005, A029837, A124767, A238279/A333755, A274174, A333381, A353832, A353849, A353863, A354584.

nn=1000;
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
q=Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,nn}];
w=Last[Select[Table[Take[q+1,i],{i,nn}],Union[#]==Range[Max@@#]&]-1];
Table[Position[w,k][[1,1]]-1,{k,0,Max@@w}]

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.

Original entry on oeis.org

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[#]&]]&]

cp's OEIS Frontend

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

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A383110 Numbers whose prime indices have a permutation with all equal run-sums.

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A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

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A353743 Least number with run-sum trajectory of length k; a(0) = 1.

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A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

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A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

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A354578 Number of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent divisors are equal.

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A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

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