A353864 -id:A353864 - cp's OEIS frontend

A353851 Number of integer compositions of n with all equal run-sums.

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 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 a(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}]

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A362607 Number of integer partitions of n with more than one mode.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 13, 13, 23, 23, 33, 45, 56, 64, 90, 101, 137, 169, 208, 246, 320, 379, 469, 567, 702, 828, 1035, 1215, 1488, 1772, 2139, 2533, 3076, 3612, 4333, 5117, 6113, 7168, 8557, 10003, 11862, 13899, 16385, 19109, 22525, 26198, 30729, 35736

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The partition (3,2,2,1,1) has greatest multiplicity 2, and two parts of multiplicity 2 (namely 1 and 2), so is counted under a(9).
The a(3) = 1 through a(9) = 9 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)
              (41)  (51)    (52)   (62)    (63)
                    (321)   (61)   (71)    (72)
                    (2211)  (421)  (431)   (81)
                                   (521)   (432)
                                   (3311)  (531)
                                           (621)
                                           (32211)
                                           (222111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)

For parts instead of multiplicities we have A002865.
For median instead of mode we have A238479, complement A238478.
These partitions have ranks A362605.
The complement is counted by A362608, ranks A356862.
For co-mode we have A362609, ranks A362606.
For co-mode complement we have A362610, ranks A359178.
A000041 counts integer partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A098859, A237984, A275870, A304442, A327472, A353864, A353865, A360071, A362612.

b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..51);  # Alois P. Heinz, May 05 2024

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]>1&]],{n,0,30}]

G_x(N)={my(x='x+O('x^(N-1)), Ib(k,j) = if(k>j,1,0), A_x(u)=sum(i=1,N-u, sum(j=u+1, N-u, (x^(i*(u+j))*(1-x^u)*(1-x^j))/((1-x^(u*i))*(1-x^(j*i))) * prod(k=1,N-i*(u+j), (1-x^(k*(i+Ib(k,j))))/(1-x^k)))));
concat([0,0,0],Vec(sum(u=1,N, A_x(u))))}
G_x(51) \\ John Tyler Rascoe, Apr 05 2024

G.f.: Sum_{u>0} A(u,x) where A(u,x) = Sum_{i>0} Sum_{j>u} ( x^(i*(u+j))*(1-x^u)*(1-x^j) )/( (1-x^(u*i))*(1-x^(j*i)) ) * Product_{k>0} ( (1-x^(k*(i+[k>j])))/(1-x^k) ) is the g.f. for partitions of this kind with least mode u and [] is the Iverson bracket. - John Tyler Rascoe, Apr 05 2024

A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154

Offset: 1

Gus Wiseman, May 05 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The prime indices of 180 are {1,1,2,2,3}, with modes {1,2}, so 180 is in the sequence, and the sequence differs from A182853.
The terms together with their prime indices begin:
     6: {1,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    21: {2,4}
    22: {1,5}
    26: {1,6}
    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}
    46: {1,9}
    51: {2,7}
    55: {3,5}

Amiram Eldar, Table of n, a(n) for n = 1..10000

The first term with bigomega n appears to be A166023(n).
The complement is A356862, counted by A362608.
For co-mode complement we have A359178, counted by A362610.
For co-mode we have A362606, counted by A362609.
Partitions of this type are counted by A362607.
These are the positions of terms > 1 in A362611.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes, ranks A362611.
A362615 counts partitions by number of co-modes, ranks A362613.
Cf. A002865, A215366, A327473, A327476, A353864, A359908, A362612.

q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
select(q, [$1..250])[];  # Alois P. Heinz, May 10 2023

Select[Range[100],Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]

is(n) = {my(e = factor(n)[, 2]); if(#e < 2, 0, e = vecsort(e); e[#e-1] == e[#e]);} \\ Amiram Eldar, Jan 20 2024

from sympy import factorint
def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
print([k for k in range(155) if ok(k)]) # Michael S. Branicky, May 06 2023

A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

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.
The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.

			Triangle begins:
   1
   2
   3
   4  3
   5
   6
   7
   8  5
   9  7
  10
  11
  12  9  7
Row 87780 is the following trajectory (left column), 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 version for run-lengths instead of sums is A325239 or A325277.
This is the iteration of A353832, with composition version A353847.
Row-lengths are A353841, counted by A353846.
Final terms are A353842.
Counting rows by final omega gives A353843.
Rows ending in a prime number are A353844, counted by A353845.
These sequences for compositions are A353853-A353859.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A182850 or A323014 gives frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
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.
A353862 gives greatest run-sum of prime indices, least A353931.
Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]

A362609 Number of integer partitions of n with more than one part of least multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

These are partitions where no part appears fewer times than all of the others.

			The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (42111)
                                             (222111)
                                             (321111)

For parts instead of multiplicities we have A117989, ranks A283050.
For median instead of co-mode we have A238479, complement A238478.
These partitions have ranks A362606.
For mode instead of co-mode we have A362607, ranks A362605.
For mode complement instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362610, ranks A359178.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]>1&]],{n,0,30}]

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]]&]

A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A130092 in lacking 180.
First differs from A351295 in lacking 180 and having 216.
First differs from A362605 in having 60.

			The prime factorization of 1800 is {2,2,2,3,3,5,5}, and the parts of least multiplicity are {3,5}, so 1800 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   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}

The complement is A359178, counted by A362610.
For mode we have A362605, counted by A362607.
Partitions of this type are counted by A362609.
These are the positions of terms > 1 in A362613.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes.
A362615 counts partitions by number of co-modes.
Cf. A215366, A327473, A327476, A353864, A356862, A359908, A362611.

Select[Range[100],Count[Last/@FactorInteger[#],Min@@Last/@FactorInteger[#]]>1&]

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

cp's OEIS Frontend

A353851 Number of integer compositions of n with all equal run-sums.

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A362607 Number of integer partitions of n with more than one mode.

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A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

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A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

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A362609 Number of integer partitions of n with more than one part of least multiplicity.

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

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

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

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