A357865 -id:A357865 - cp's OEIS frontend

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

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[#]]&]

A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 9, 8, 11, 11, 20, 16, 20, 21, 32, 30, 41, 38, 50, 48, 62, 64, 89, 81, 97, 100, 123, 123, 151, 154, 187, 183, 221, 221, 279, 272, 312, 316, 377, 376, 446, 460, 531, 547, 628, 641, 754, 746, 841, 856, 990, 1007, 1145, 1167, 1325, 1346, 1519, 1567, 1776

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly increasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6

Cf. A100882.
These partitions are ranked by A357861.
The complement is A357865, ranked by A357850.
The opposite version is A304405, ranked by A357875.
The strict version is A304430, ranked by A357864.
The strict opposite version is A304428, ranked by A357862.
Number of rows in A354584 summing to n that are weakly decreasing.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A239312, A275870, A353832, A353833, A353864.

Table[Length[Select[IntegerPartitions[n],LessEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 40, 41, 43, 45, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157

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:
    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}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.

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

These partitions are counted by A304406.
These are the indices  of rows in A354584 that are weakly decreasing.
The complement is A357850, counted by A357865, opposite A357876.
The strictly decreasing version is A357864, counted by A304430.
The opposite (weakly increasing) version is A357875, counted by A304405.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A300273, A304442.

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

cp's OEIS Frontend

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

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A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

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A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

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

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