A287170 -id:A287170 - cp's OEIS frontend

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

1, 2, 2, 3, 2, 3, 2, 5, 3, 4, 2, 5, 2, 4, 3, 7, 2, 5, 2, 6, 4, 4, 2, 7, 3, 4, 5, 6, 2, 5, 2, 11, 4, 4, 3, 7, 2, 4, 4, 10, 2, 6, 2, 6, 5, 4, 2, 11, 3, 6, 4, 6, 2, 7, 4, 10, 4, 4, 2, 7, 2, 4, 6, 13, 4, 6, 2, 6, 4, 6, 2, 11, 2, 4, 5, 6, 3, 6, 2, 14, 7, 4, 2, 10

Offset: 1

Gus Wiseman, Aug 18 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.
A multiset is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), with Heinz number 30, so a(18564) = 30.

Positions of prime terms are A073491, complement A073492.
Positions of terms with bigomega 2-4 are A073493-A073495.
Applying bigomega gives A287170, firsts A066205, even bisection A356229.
These are the Heinz numbers of the rows of A356226.
Minimal/maximal prime indices are A356227/A356228.
A version for standard compositions is A356230, firsts A356232/A356603.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A001222(a(n)) = A287170(n).
A055396(a(n)) = A356227(n).
A061395(a(n)) = A356228(n).

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 4, 5, 2, 6, 2, 4, 3, 3, 2, 8, 3, 3, 4, 4, 2, 7, 2, 6, 3, 3, 4, 9, 2, 3, 3, 5, 2, 5, 2, 4, 6, 3, 2, 10, 3, 4, 3, 4, 2, 8, 3, 5, 3, 3, 2, 10, 2, 3, 4, 7, 3, 5, 2, 4, 3, 5, 2, 12, 2, 3, 6, 4, 4, 5, 2, 6, 5, 3, 2, 7, 3, 3

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

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 a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

These divisors belong to A073491, a superset of A055932, complement A073492.
The initial case is A356224.
The complement in the initial case is counted by A356225.
A000005 counts divisors.
A001223 lists the prime gaps.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8

Offset: 1

Gus Wiseman, Aug 28 2022

Row-products are the positive integers 1, 2, 3, ...

			The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Row-lengths are A287170, firsts A066205, even bisection A356229.
Applying bigomega to all parts gives A356226, statistics A356227-A356232.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A356228 Greatest size of a gapless submultiset of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 5, 2, 2, 1, 2, 1, 4, 1, 3, 1, 1, 1, 4, 1, 1, 2, 6, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 3, 2, 2, 2, 1, 4, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless intervals: {2,3}, {5,5,6}, {9}.

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 prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-maxima of A356226, firsts A356232.
The smallest instead of greatest size is A356227.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356069 counts gapless divisors.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Max@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333766(A356230(n)).

a(n) = A061395(A356231(n)).

A356229 Number of maximal gapless submultisets of the prime indices of 2n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Gus Wiseman, Aug 16 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
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.
This is a bisection of A287170, but is important in its own right because the even numbers are exactly those whose prime indices begin with 1.

			The prime indices of 2*9282 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(9282) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

This is the even (bisected) case of A287170, firsts A066205.
Alternate row-lengths of A356226, minima A356227(2n), maxima A356228(2n).
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime indices, listed by A112798, sum A056239.
A003963 multiplies together the prime indices of n.
A073093 counts the prime indices of 2n.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
Cf. A000005, A060680, A060681, A132747, A132881, A286470, A289509, A356230, A356231, A356232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[primeMS[2n],#1>=#2-1&]],{n,100}]

A287170(n) = { my(f=factor(n)); if(#f~==0, return (0), return(#f~ - sum(i=1, #f~-1, if (primepi(f[i, 1])+1 == primepi(f[i+1, 1]), 1, 0)))); };
A356229(n) = A287170(2*n); \\ Antti Karttunen, Jan 19 2025

a(n) = A287170(2n).

Data section extended to a(105) by Antti Karttunen, Jan 19 2025

A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

1, 2, 4, 10, 8, 20, 50, 110, 16, 40, 100, 220, 250, 550, 1210, 1870, 32, 80, 200, 440, 500, 1100, 2420, 3740, 1250, 2750, 6050, 9350, 13310, 20570, 31790, 43010, 64, 160, 400, 880, 1000, 2200, 4840, 7480, 2500, 5500, 12100, 18700, 26620, 41140, 63580, 86020

Offset: 0

Gus Wiseman, Aug 30 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.

The image consists of all numbers whose prime indices are odd and cover an initial interval of odd positive integers.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     10: {1,3}
      8: {1,1,1}
     20: {1,1,3}
     50: {1,3,3}
    110: {1,3,5}
     16: {1,1,1,1}
     40: {1,1,1,3}
    100: {1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    550: {1,3,3,5}
   1210: {1,3,5,5}
   1870: {1,3,5,7}

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The partitions with these Heinz numbers are counted by A053251.
A subset of A066208 (numbers with all odd prime indices).
Up to permutation, these are the positions of first appearances of rows in A356226. Other statistics are:
- length: A287170, firsts A066205
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
The sorted version is A356232.
An ordered version is counted by A356604.
A001221 counts distinct prime factors, sum A001414.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A001222, A055932, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=1/2 Total[2^Accumulate[Reverse[q]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
sq=stcinv/@Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,1000}];
Table[Position[sq,k][[1,1]],{k,0,mnrm[Rest[sq]]}]

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 1, 1, 1, 1, 4, 2, 1, 3, 1, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 1.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-minima of A356226, firsts A356232.
The greatest instead of smallest size is A356228.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).

A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 13, 15, 18, 22, 28, 31, 38, 45, 53, 62, 74, 86, 105, 123, 146, 171, 208, 242, 290, 340, 399, 469, 552, 639, 747, 862, 999, 1150, 1326, 1514, 1736, 1979, 2256, 2560, 2909, 3283, 3721, 4191, 4726, 5311, 5973, 6691, 7510, 8396, 9395

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
                 (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
                        (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
                                 (4,1,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (4,2,2)  (4,4,1)
                                          (5,1,1)  (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                   (6,1,1)  (6,2,1)
                                                            (7,1,1)

For subsets instead of strict partitions we have A384177, for runs A384175.
The strict case is A384880.
For runs instead of anti-runs we have A384884, strict A384178.
For equal instead of distinct lengths we have A384888, for runs A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]

A384881 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive parts decreasing by 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 2, 2, 2, 0, 1, 0, 2, 3, 3, 2, 0, 1, 0, 2, 5, 3, 2, 2, 0, 1, 0, 1, 8, 4, 4, 2, 2, 0, 1, 0, 3, 5, 10, 4, 3, 2, 2, 0, 1, 0, 2, 9, 9, 9, 5, 3, 2, 2, 0, 1, 0, 2, 11, 13, 9, 9, 4, 3, 2, 2, 0, 1

Offset: 0

Gus Wiseman, Jun 25 2025

			The partition (5,4,2,1,1) has maximal runs ((5,4),(2,1),(1)) so is counted under T(13,3) = 23.
Row n = 9 counts the following partitions:
  9    63    333    6111    33111   411111   3111111   111111111
  54   72    441    22221   51111   2211111  21111111
  432  81    522    42111   222111
       621   531    321111
       3321  711
             3222
             4221
             4311
             5211
             32211
Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  1  3  0  1
  0  2  2  2  0  1
  0  2  3  3  2  0  1
  0  2  5  3  2  2  0  1
  0  1  8  4  4  2  2  0  1
  0  3  5 10  4  3  2  2  0  1
  0  2  9  9  9  5  3  2  2  0  1
  0  2 11 13  9  9  4  3  2  2  0  1
  0  2 13 15 17  8 10  4  3  2  2  0  1
  0  2 14 23 16 17  8  9  4  3  2  2  0  1
  0  2 16 26 26 19 16  9  9  4  3  2  2  0  1
  0  4 13 37 32 26 19 16  8  9  4  3  2  2  0  1

John Tyler Rascoe, Rows n = 0..100, flattened

Row sums are A000041.
Column k = 1 is A001227.
For distinct parts instead of maximal runs we have A116608.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
Partitions with distinct runs of this type are counted by A384882, gapless A384884.
For prime indices see A385213, A287170, A066205, A356229.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A000217, A008284, A047966, A047993, A098859, A325325, A375136, A384887.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1==#2+1&]]==k&]],{n,0,10},{k,0,n}]

tri(n) = {(n*(n+1)/2)}
B_list(N) = {my(v = vector(N, i, 0)); v[1] = q*t; for(m=2,N, v[m] = t * (q^tri(m) + sum(i=1,m-1, q^tri(i) * v[m-i] * (q^((m-i)*(i-1))/(1 - q^(m-i)) - q^((m-i)*i) + O('q^(N-tri(i)+1)))))); v}
A_qt(max_row) = {my(N = max_row+1, B = B_list(N), g = 1 + sum(m=1,N, B[m]/(1 - q^m)) + O('q^(N+1))); vector(N, n, Vecrev(polcoeff(g, n-1)))} \\ John Tyler Rascoe, Aug 18 2025

G.f.: 1 + Sum_{m>0} B(m,q,t)/(1 - q^m) where B(m,q,t) = t * (q^tri(m) + Sum_{i=1..m-1} q^tri(i) * B(m-i,q,t) * ((q^((m-i)*(i-1))/(1 - q^(m-i))) - q^((m-i)*i))) and tri(n) = A000217(n). - John Tyler Rascoe, Aug 18 2025

A356845 Odd numbers with gapless prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Gus Wiseman, Sep 03 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 sequence is gapless if it covers an interval of positive integers.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Consists of the odd terms of A073491.
These partitions are counted by A264396.
The strict case is A294674, counted by A136107.
The version for compositions is A356843, counted by A251729.
A001221 counts distinct prime factors, sum A001414.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[1,100,2],nogapQ[primeMS[#]]&]

cp's OEIS Frontend

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

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A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

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A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

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A356228 Greatest size of a gapless submultiset of the prime indices of n.

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A356229 Number of maximal gapless submultisets of the prime indices of 2n.

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A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).

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A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

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A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

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