A353866 -id:A353866 - cp's OEIS frontend

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

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

A353834 Nonprime numbers whose prime indices have all equal run-sums.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401

Offset: 1

Gus Wiseman, May 26 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:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    40: {1,1,1,3}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   112: {1,1,1,1,4}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).

For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
For compositions we have A353848\A000079, counted by A353851(n)-1.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
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. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.

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

from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue,1)))
A353848_list = list(islice(A353848_gen(),30)) # Chai Wah Wu, May 27 2022

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

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@@#&]]&]

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 3, 4, 3, 2, 4, 3, 3, 4, 4, 3, 4, 3, 4, 5, 5, 4, 6, 4, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 9, 6, 6, 7, 6, 8, 9, 6, 6, 8, 9, 7, 9, 9, 7, 10, 9, 8, 13, 7, 10, 11, 8, 9, 10, 11, 12, 9, 11, 9, 15, 12, 12, 19, 13, 16, 16

Offset: 0

Gus Wiseman, Jun 04 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). A weak run-sum is the sum of any consecutive constant subsequence.

Do all positive integers appear only finitely many times in this sequence?

			The a(n) compositions for n = 1, 3, 9, 15, 18:
  (1)  (21)   (4311)       (54321)            (543321)
       (111)  (51111)      (532221)           (654111)
              (111111111)  (651111)           (7611111)
                           (81111111)         (111111111111111111)
                           (111111111111111)
For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).

Perfect partitions are counted by A002033, ranked by A325780.
Knapsack partitions are counted by A108917, ranked by A299702.
This is the complete case of A353864, ranked by A353866.
These partitions are ranked by A353867.
A000041 counts partitions, strict A000009.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353850 counts compositions with all distinct run-sums, ranked by A353852.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.

norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]

a(n) = my(c=0, s, v); if(n, forpart(p=n, if(p[1]==1, v=List([s=1]); for(i=2, #p, if(p[i]==p[i-1], listput(v, s+=p[i]), listput(v, s=p[i]))); s=#v; listsort(v, 1); if(s==#v&&s==v[s], c++))); c, 1); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 2, 5, 0, 0, 0, 0, 5, 5, 1, 0, 0, 0, 0, 2, 12, 1, 0, 0, 0, 0, 0, 7, 12, 3, 0, 0, 0, 0, 0, 0, 3, 19, 8, 0, 0, 0, 0, 0, 0, 0, 5, 27, 9, 1, 0, 0, 0, 0, 0, 0, 0, 2, 33, 20, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, May 26 2022

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  4  1  0  0
  0  2  5  0  0  0
  0  5  5  1  0  0  0
  0  2 12  1  0  0  0  0
  0  7 12  3  0  0  0  0  0
  0  3 19  8  0  0  0  0  0  0
  0  5 27  9  1  0  0  0  0  0  0
  0  2 33 20  1  0  0  0  0  0  0  0
  0 13 28 34  2  0  0  0  0  0  0  0  0
  0  2 48 46  5  0  0  0  0  0  0  0  0  0
  0  5 65 51 14  0  0  0  0  0  0  0  0  0  0
  0  4 57 99 15  1  0  0  0  0  0  0  0  0  0  0
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (41111)     (611)
  (221111)    (3311)
  (11111111)  (4211)
              (5111)
              (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Counting distinct parts instead of run-sums gives A116608.
Column k = 1 is A304442, ranked by A353833 (nonprime A353834).
The rank statistic is A353835, weak A353861, for compositions A353849.
A275870 counts collapsible partitions, ranked by A300273.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
Cf. A008284, A047966, A071625, A165413, A175413, A325280, A333755.

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

A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 25 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.
Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.
Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.

			The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
  1080: {1,1,1,2,2,2,3}
   325: {3,3,6}
   169: {6,6}
    37: {12}
The trajectory for a(87780) = 5 is the following, with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}
The trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
  39960: {1,1,1,2,2,2,3,12}
  12025: {3,3,6,12}
   6253: {6,6,12}
   1369: {12,12}
     89: {24}

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 1's are A005117.
The version for run-lengths instead of sums is A182850 or A323014.
Positions of first appearances are A353743.
These are the row-lengths of A353840.
Other sequences pertaining to this trajectory are A353842-A353845.
Counting partitions by this statistic gives A353846.
The version for compositions is A353854, run-lengths of A353853.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A318928 gives runs-resistance of binary expansion.
A353832 represents the operation of taking run-sums of a partition.
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. A008966, A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.

Table[If[n==1,0,Length[NestWhileList[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&]]],{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)));
A353841(n) = if(1==n,0,for(i=1,oo,if(issquarefree(n), return(i), n = A353832(n)))); \\ Antti Karttunen, Jan 20 2025

a(1) = 0, and for n > 1, if A008966(n) = 1 [n is in A005117], a(n) = 1, otherwise a(n) = 1+a(A353832(n)). [See comments] - Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

A353931 Least run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 1, 8, 2, 2, 1, 9, 2, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 3, 13, 1, 14, 2, 3, 1, 15, 2, 8, 1, 2, 2, 16, 1, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 3, 21, 1, 2, 2, 4, 1, 22, 3, 8, 1

Offset: 1

Gus Wiseman, Jun 07 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 prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

Positions of first appearances are A008578.
For run-lengths instead of run-sums we have A051904, greatest A051903.
For run-sums and binary expansion we have A144790, greatest A038374.
For run-lengths and binary expansion we have A175597, greatest A043276.
Distinct run-sums are counted by A353835, weak A353861.
The greatest run-sum is given by A353862.
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.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A353862 Greatest run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 2, 6, 4, 3, 4, 7, 4, 8, 3, 4, 5, 9, 3, 6, 6, 6, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 3, 13, 4, 14, 5, 4, 9, 15, 4, 8, 6, 7, 6, 16, 6, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 4, 21, 12, 6, 8, 5, 6, 22, 4, 8

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 run-sum of a sequence is the sum of any maximal consecutive constant subsequence.

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 4.

Positions of first appearances are A008578.
For binary expansion we have A038374, least A144790.
For run-lengths instead of run-sums we have A051903.
Distinct run-sums are counted by A353835, weak A353861.
The least run-sum is given by 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.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A182857, A316413, A353836, A353839, A353848, A353852, A353866.

Table[Max@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 19 2025

nonn

Also the number of factorizations of n into prime powers > 1 with equal sums of prime indices.

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 2 multiset partitions into constant blocks with a common sum:
  {{2,2},{1,1,1,1}}
  {{2},{2},{1,1},{1,1}}
so a(144) = 2.

For just constant blocks we have A000688.
Twice-partitions of this type are counted by A279789.
For just a common sum we have A321455.
For distinct instead of equal sums we have A381635.
Positions of 0 are A381871, counted by A381993.
MM-numbers of these multiset partitions are A382215.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A279784, A295935, A381453 (lower), A381455 (upper).
Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.
Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@SameQ@@@#&]],{n,100}]

A323774(n) = Sum_{A056239(k)=n} a(k). Gus Wiseman, Apr 25 2025

cp's OEIS Frontend

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

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A353834 Nonprime numbers whose prime indices have all equal run-sums.

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A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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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.

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A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

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A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

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A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

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A353931 Least run-sum of the prime indices of n.

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