A300273 -id:A300273 - cp's OEIS frontend

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

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[#]]&]

A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 43, 46, 47, 58, 59, 60, 62, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 171, 174, 175, 184, 186, 187, 190, 191, 232, 234, 235, 238, 239, 248, 250, 251, 254, 255, 256, 292, 295, 316, 319, 484

Offset: 1

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.

If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.

			The terms together with their corresponding compositions begin:
  1:(1)  2:(2)   4:(3)     8:(4)     16:(5)      32:(6)
         3:(11)  7:(111)  10:(22)    31:(11111)  36:(33)
                          11:(211)               39:(3111)
                          14:(112)               42:(222)
                          15:(1111)              43:(2211)
                                                 46:(2112)
                                                 47:(21111)
                                                 58:(1122)
                                                 59:(11211)
                                                 60:(1113)
                                                 62:(11112)
                                                 63:(111111)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for Heinz numbers of partitions is A300273, counted by A275870.
These compositions are counted by A353860.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A334968 counts distinct sums of subsequences of standard compositions.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A354582 counts distinct partial runs of standard compositions, sums A354907.
Cf. A005811, A029837, A072639, A124771, A274174, A318928, A333381, A353848, A353849, A353850.

repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[repcams[stc[#]],{_}]&]

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 11, 2, 2, 1, 33, 1, 116, 1, 5, 4, 435, 1, 2, 11, 1, 2, 1832, 2, 8167, 1, 12, 33, 10, 1, 39700, 116, 37, 1, 201785, 5, 1099449, 4, 3, 435, 6237505, 1, 19, 2, 123, 11, 37406458, 1, 27, 2, 474, 1832, 232176847, 2, 1513796040

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(21) = 5 maximal chains are the rows:
(111111)<(21111)<(2211)<(222)<(42)
(111111)<(21111)<(2211)<(411)<(42)
(111111)<(21111)<(2211)<(321)<(42)
(111111)<(21111)<(3111)<(411)<(42)
(111111)<(21111)<(3111)<(321)<(42)

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300385.

pcovs[ptn_]:=Select[Union[Reverse/@Sort/@Join@@@Tuples[IntegerPartitions/@ptn]],Length[#]===Length[ptn]+1&];
coc[ptn_]:=coc[ptn]=If[Max[ptn]===1,1,Total[coc/@pcovs[ptn]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[coc[Reverse[primeMS[n]]],{n,50}]

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}]

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Gus Wiseman, Apr 17 2025

nonn

First differs from A381871 in having 36.

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 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
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}
   24: {1,1,1,2}
   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}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}

For run-lengths instead of sums we have A059404, distinct A130092.
The complement is A353833, counted by A304442.
For distinct instead of equal run-sums we have A353839.
Partitions of this type are counted by A382076.
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)
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with a common run-sum, ranks A353848.
A353862 gives the greatest run-sum of prime indices, least A353931.
A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

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

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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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A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

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A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

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

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A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

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