A354584 -id:A354584 - cp's OEIS frontend

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.

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}]

A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 23 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).

			Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), with distinct sums {1,2,3,4,6}, so a(981) = 5.

Positions of 1's are A000051.
Positions of first appearances are A000079.
The standard compositions used here are A066099, run-sums A353847/A353932.
If we allow any subsequence we get A334968.
The case of full runs is A353849, firsts A246534.
A version for nonempty partitions is A353861, full A353835.
Counting all distinct runs (instead of their distinct sums) gives A354582.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A330036 counts distinct partial runs of prime indices, full A005811.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A354584 lists run-sums of prime indices, rows ranked by A353832.
Cf. A029837, A124771, A274174, A333381, A334299, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
Table[Length[Union[Total/@Join@@pre/@Split[stc[n]]]],{n,0,100}]

A357876 The run-sums of the prime indices of n are not weakly increasing.

Original entry on oeis.org

24, 45, 48, 80, 90, 96, 120, 135, 160, 168, 175, 180, 189, 192, 224, 240, 264, 270, 275, 288, 297, 312, 315, 320, 336, 350, 360, 378, 384, 405, 408, 448, 456, 480, 495, 525, 528, 539, 540, 550, 552, 560, 567, 576, 585, 594, 600, 624, 630, 637, 640, 672, 696

Offset: 1

Gus Wiseman, Oct 17 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 terms together with their prime indices begin:
   24: {1,1,1,2}
   45: {2,2,3}
   48: {1,1,1,1,2}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  135: {2,2,2,3}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
  175: {3,3,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  192: {1,1,1,1,1,1,2}
For example, the prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is in the sequence.

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

These are the indices  of rows in A354584 that are not weakly increasing.
The complement is A357875.
These partitions are counted by A357878.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.

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

A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 40, 55, 79, 104, 150, 196, 270, 350, 467, 600, 786, 997, 1293, 1632, 2077, 2597, 3283, 4067, 5088, 6268, 7769, 9517, 11704, 14238, 17405, 21092, 25598, 30861, 37278, 44729, 53742, 64226, 76811, 91448, 108929, 129174

Offset: 0

Gus Wiseman, Oct 18 2022

nonn

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 a(0) = 0 through a(9) = 8 partitions:
  .  .  .  .  .  (2111)  (21111)  (322)     (3221)     (3222)
                                  (31111)   (32111)    (32211)
                                  (211111)  (311111)   (42111)
                                            (2111111)  (321111)
                                                       (411111)
                                                       (2211111)
                                                       (3111111)
                                                       (21111111)

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

The complement is counted by A304405, ranked by A357875.
Number of rows in A354584 summing to n that are weakly increasing.
The opposite (not weakly increasing) version is A357865, ranked by A357850.
These partitions are ranked by A357876.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A098859, A239312, A275870, A304406, A304428, A304430, A353832, A353864, A353932.

Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[Reverse[#]]&]],{n,0,30}]

A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111

Offset: 1

Gus Wiseman, Oct 19 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 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 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 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}
   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}

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

These are the indices of rows in A354584 that are not weakly decreasing.
The complement is A357861, counted by A304406.
These partitions are counted by A357865.
The opposite (not weakly increasing) version is A357876, counted by A357878.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A118914, A181819, A300273, A304405, A304428, A304430, A304442, A353832, A353864, A353932, A357864, A357875.

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

A357865 Number of integer partitions of n whose run-sums are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 5, 10, 13, 22, 31, 45, 57, 85, 115, 155, 199, 267, 344, 452, 577, 744, 940, 1191, 1486, 1877, 2339, 2910, 3595, 4442, 5453, 6688, 8162, 9960, 12089, 14662, 17698, 21365, 25703, 30869, 36961, 44207, 52728, 62801, 74644, 88587, 104930, 124113

Offset: 0

Gus Wiseman, Oct 19 2022

nonn

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 a(0) = 0 through a(8) = 13 partitions:
  .  .  .  (21)  (31)  (32)   (42)    (43)     (53)
                       (41)   (51)    (52)     (62)
                       (221)  (321)   (61)     (71)
                       (311)  (411)   (331)    (332)
                              (2211)  (421)    (431)
                                      (511)    (521)
                                      (2221)   (611)
                                      (3211)   (3221)
                                      (4111)   (3311)
                                      (22111)  (4211)
                                               (5111)
                                               (22211)
                                               (32111)

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

The complement is counted by A304406, ranked by A357861.
Number of rows in A354584 summing to n that are not weakly decreasing.
These partitions are ranked by A357850.
The opposite (not weakly decreasing) version is A357878, ranked by A357876.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A098859, A239312, A275870, A304405, A304428, A304430, A353832, A353864, A357875.

Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[#]&]],{n,0,30}]

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]&]

A357863 Numbers whose prime indices do not have strictly increasing run-sums. Heinz numbers of the partitions not counted by A304428.

Original entry on oeis.org

12, 24, 40, 45, 48, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 180, 189, 192, 204, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405, 408, 420, 440

Offset: 1

Gus Wiseman, Oct 19 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 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 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 terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   40: {1,1,1,3}
   45: {2,2,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  135: {2,2,2,3}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}

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

These are the indices  of rows in A354584 that are not strictly increasing.
The complement (strictly increasing) is A357862, counted by A304428.
The weak (not weakly increasing) version is A357876, counted by A357878.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A118914, A181819, A300273, A304430, A304442, A357864, A357875.

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

A357877 The a(n)-th composition in standard order is the sequence of run-sums of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 8, 4, 8, 12, 16, 10, 32, 24, 20, 8, 64, 24, 128, 20, 40, 48, 256, 18, 32, 96, 32, 40, 512, 52, 1024, 16, 80, 192, 72, 40, 2048, 384, 160, 36, 4096, 104, 8192, 80, 68, 768, 16384, 34, 128, 96, 320, 160, 32768, 96, 144, 72, 640, 1536, 65536, 84

Offset: 1

Gus Wiseman, Oct 17 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 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 prime indices of 24 are (1,1,1,2), with run-sums (3,2), and this is the 18th composition in standard order, so a(24) = 18.

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

The version for prime indices instead of standard compositions is A353832.
The version for standard compositions instead of prime indices is A353847.
A ranking of the rows of A354584.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A056239 adds up prime indices, row sums of A112798.
A066099 lists standard compositions.
A351014 counts distinct runs in standard compositions.
Cf. A118914, A181819, A238279, A239312, A275870, A300273, A304405, A304442, A304660, A333755, A353743-A354912, A357875.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[primeMS[n]]],{n,100}]

cp's OEIS Frontend

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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A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

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A357876 The run-sums of the prime indices of n are not weakly increasing.

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A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.

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A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

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A357865 Number of integer partitions of n whose run-sums are not weakly increasing.

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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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A357863 Numbers whose prime indices do not have strictly increasing run-sums. Heinz numbers of the partitions not counted by A304428.

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