A353852 -id:A353852 - cp's OEIS frontend

A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 4, 2, 2, 0, 0, 7, 7, 2, 0, 0, 0, 14, 14, 4, 0, 0, 0, 0, 23, 29, 12, 0, 0, 0, 0, 0, 39, 56, 25, 8, 0, 0, 0, 0, 0, 71, 122, 53, 10, 0, 0, 0, 0, 0, 0, 124, 246, 126, 16, 0, 0, 0, 0, 0, 0, 0, 214, 498, 264, 48, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 02 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 run-sum trajectory is obtained by repeatedly taking the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,4,2,1,1) -> (2,4,2,2) -> (2,4,4) -> (2,8) is counted under T(10,4).

			Triangle begins:
   1
   0   1
   0   1   1
   0   3   1   0
   0   4   2   2   0
   0   7   7   2   0   0
   0  14  14   4   0   0   0
   0  23  29  12   0   0   0   0
   0  39  56  25   8   0   0   0   0
   0  71 122  53  10   0   0   0   0   0
   0 124 246 126  16   0   0   0   0   0   0
   0 214 498 264  48   0   0   0   0   0   0   0
For example, row n = 5 counts the following compositions:
  (5)    (113)    (1121)
  (14)   (122)    (1211)
  (23)   (221)
  (32)   (311)
  (41)   (1112)
  (131)  (2111)
  (212)  (11111)

Column k = 1 is A003242, ranked by A333489, complement A261983.
Row sums are A011782.
Positive row-lengths are A070939.
The version for partitions is A353846, ranked by A353841.
This statistic (trajectory length) is ranked by A353854, firsts A072639.
Counting by length of last part instead of number of parts gives A353856.
A333627 ranks the run-lengths of standard compositions.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.

rsc[y_]:=If[y=={},{},NestWhileList[Total/@Split[#]&,y,MatchQ[#,{_,x_,x_,_}]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[rsc[#]]==k&]],{n,0,10},{k,0,n}]

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 4, 3, 4, 1, 3, 5, 2, 2, 6, 1, 4, 2, 3, 4, 7, 1, 4, 8, 2, 3, 2, 4, 1, 5, 9, 3, 2, 6, 1, 6, 6, 2, 4, 10, 1, 2, 3, 11, 5, 2, 5, 1, 7, 3, 4, 2, 4, 12, 1, 8, 2, 6, 3, 3, 13, 1, 2, 4, 14, 2, 5, 4, 3, 1, 9, 15, 4, 2, 8, 1, 6, 2, 7, 2, 6, 16

Offset: 1

Gus Wiseman, Jun 17 2022

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.

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).

			Triangle begins:
  .
  1
  2
  2
  3
  1 2
  4
  3
  4
  1 3
  5
  2 2
  6
  1 4
  2 3
For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).

Positions of first appearances are A308495 plus 1.
The version for compositions is A353932, ranked by A353847.
Classes:
- singleton rows: A000961
- constant rows: A353833, nonprime A353834, counted by A304442
- strict rows: A353838, counted by A353837, complement A353839
Statistics:
- row lengths: A001221
- row sums: A056239
- row products: A304117
- row ranks (as partitions): A353832
- row image sizes: A353835
- row maxima: A353862
- row minima: A353931
A001222 counts prime factors with multiplicity.
A112798 and A296150 list partitions by rank.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct sums of partial runs of prime indices.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.

Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,30}]

A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Gus Wiseman, Jun 06 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.

In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
The sequence contains 18 because its prime-power divisors {1,2,3,9} have prime indices {}, {1}, {2}, {2,2} with distinct sums {0,1,2,4}. On the other hand, 12 is not in the sequence because {2} and {1,1} have the same sum.

Knapsack partitions are counted by A108917, ranked by A299702.
The strong case is A353838, counted by A353837, complement A353839.
These partitions are counted by A353864.
The complete case is A353867, counted by A353865.
The complement is A354583.
A000041 counts partitions, strict A000009.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.

msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Select[msubs[primeMS[#]],SameQ@@#&]&]

A353839 Numbers whose prime indices do not have all distinct run-sums.

Original entry on oeis.org

12, 40, 60, 63, 84, 112, 120, 126, 132, 144, 156, 204, 228, 252, 276, 280, 300, 315, 325, 336, 348, 351, 352, 360, 372, 420, 440, 444, 492, 504, 516, 520, 560, 564, 588, 630, 636, 650, 660, 675, 680, 693, 702, 708, 720, 732, 760, 780, 804, 819, 832, 840, 852

Offset: 1

Gus Wiseman, Jun 04 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.

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 prime indices begin:
   12: {1,1,2}
   40: {1,1,1,3}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  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}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  280: {1,1,1,3,4}
  300: {1,1,2,3,3}
  315: {2,2,3,4}

For equal run-sums we have A353833, counted by A304442, nonprime A353834.
The complement is A353838, counted by A353837.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353862 gives the greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002110, A071625, A073093, A116608, A118914, A124010, A353861, A353867.

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

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

A353860 Number of collapsible integer compositions of n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 12, 2, 26, 9, 36, 2, 206, 2, 132, 40, 677, 2, 1746, 2, 3398, 136, 2052, 2, 44388, 33, 8196, 730, 79166, 2, 263234, 2, 458330, 2056, 131076, 160, 8804349, 2, 524292, 8200, 13662156, 2, 36036674, 2, 48844526, 90282, 8388612, 2, 1971667502, 129

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

If a collapse is a joining of some number of adjacent equal parts of an integer composition, we call a composition collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (1,1,1,3,2,1,1,2) -> (3,3,2,1,1,2) -> (3,3,2,2,2) -> (6,2,2,2) -> (6,6) -> (12), which shows that (1,1,1,3,2,1,1,2) is a collapsible composition of 12.

			The a(0) = 0 through a(6) = 12 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)
          (11)  (111)  (22)    (11111)  (33)
                       (112)            (222)
                       (211)            (1113)
                       (1111)           (1122)
                                        (2112)
                                        (2211)
                                        (3111)
                                        (11112)
                                        (11211)
                                        (21111)
                                        (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for partitions is A275870, ranked by A300273.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A318928, A333755, A353844, A353848, A353849, A353850, A353852.

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,#]]&]]]];
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[repcams[#],{n}]&]],{n,0,15}]

a(n) = if(n==0, 0, 1 - sumdiv(n, d, if(d>1, moebius(d)*a(n/d)^d ))) \\ Andrew Howroyd, Feb 04 2023

Sum_{d|n} mu(d)*a(n/d)^d = 1 for n > 0. - Andrew Howroyd, Feb 04 2023

Terms a(16) and beyond from Andrew Howroyd, Feb 04 2023

A353744 Numbers k such that the k-th composition in standard order has all equal run-lengths.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 52, 54, 58, 63, 64, 65, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 101, 102, 104, 105, 108, 109, 127, 128

Offset: 1

Gus Wiseman, Jun 11 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.

			Composition 2362 in standard order is (3,3,1,1,2,2), with run-lengths (2,2,2), so 2362 is in the sequence.

Standard compositions are listed by A066099.
The version for partitions is A072774, counted by A047966.
These compositions are counted by A329738.
For distinct instead of equal run-lengths we have A351596.
For run-sums instead of lengths we have A353848, counted by A353851.
For distinct run-sums we have A353852, counted by A353850.
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.
A353847 represents the composition run-sum transformation.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353833 ranks partitions with all equal run-sums, counted by A304442.
Cf. A029837, A071625, A124767, A175413, A238279, A333381, A333755, A353834, A353849.

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

A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 0, 1, 2, 0, 1, 2, 6, 1, 2, 2, 6, 1, 4, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 12 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.

A run in a sequence is a constant consecutive subsequence. The run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums. See A353932 for run-sums of standard compositions.

			For n = 12, none of the permutations (1,1,2), (1,2,1), (2,1,1) has distinct run-sums, so a(12) = 0.
The prime indices of 36 are {1,1,2,2}, and we have permutations: (1,1,2,2), (2,2,1,1), so a(36) = 2.
For n = 90 we have:
  (1,2,2,3)
  (1,3,2,2)
  (2,2,1,3)
  (2,2,3,1)
  (3,1,2,2)
  (3,2,2,1)
So a(90) = 6. The 6 missing permutations are: (1,2,3,2), (2,1,2,3), (2,1,3,2), (2,3,1,2), (2,3,2,1), (3,2,1,2).

Positions of 1 are A000961.
Compositions of this type are counted by A353850, ranked by A353852.
Positions of 0 appear to be A381636, for equal run-sums A383100.
For run-lengths instead of sums we have A382771, equal A382857 (zeros A382879).
For equal instead of distinct run-sums we have A382877.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A001221, A001222, A098859, A130091, A329738, A351013, A351202, A353848, A353851, A354580, A354584.

Table[Length[Select[Permutations[PrimePi /@ Join@@ConstantArray@@@FactorInteger[n]], UnsameQ@@Total/@Split[#]&]],{n,100}]

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Gus Wiseman, Jun 07 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.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}

Knapsack partitions are counted by A108917, ranked by A299702.
Complete partitions are counted by A126796, ranked by A325781.
These partitions are counted by A353865.
This is a special case of A353866, counted by A353864, complement A354583.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

cp's OEIS Frontend

A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353839 Numbers whose prime indices do not have all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353861 Number of distinct weak run-sums of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A353860 Number of collapsible integer compositions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A353744 Numbers k such that the k-th composition in standard order has all equal run-lengths.

Views

Author

Keywords

Comments