A356227 -id:A356227 - cp's OEIS frontend

A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Rémy Sigrist, Jun 04 2017

nonn

a(n) = 0 iff n = 1.
a(n) = 1 iff n belongs to A073491.
a(p) = 1 for any prime p.
a(A002110(n)) = 1 for any n > 0.
a(n!) = 1 for any n > 1.
a(A066205(n)) = n for any n > 0.
a(n) = a(A007947(n)) for any n > 0.
a(n) = a(A003961(n)) for any n > 0.
a(n*m) <= a(n) + a(m) for any n > 0 and m > 0.
Each number n can be uniquely represented as a product of a(n) distinct terms from A073491; this representation is minimal relative to the number of terms.

			See illustration of the first terms in the Links section.
The prime indices of 18564 are {1,1,2,4,6,7}, which separate into maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3; this corresponds to the ordered factorization 18564 = 12 * 7 * 221. - _Gus Wiseman_, Sep 03 2022

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Cf. A002110, A003961, A007947, A069010, A087207.
Positions of first appearances are A066205.
These are the row-lengths of A356226 and A356234. Other statistics are:
- length: A287170 (this sequence)
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356603 or A356232 (sorted)
A001222 counts prime factors, distinct A001221.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A053251, A055932, A061395, A073493, A107428, A132747, A137921, A193829, A286470.

Table[Length[Select[First/@If[n==1,{},FactorInteger[n]],!Divisible[n,NextPrime[#]]&]],{n,30}] (* Gus Wiseman, Sep 03 2022 *)

a(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))))

from sympy import factorint, primepi
def a087207(n):
    f=factorint(n)
    return sum([2**primepi(i - 1) for i in f])
def a069010(n): return sum(1 for d in bin(n)[2:].split('0') if len(d)) # this function from Chai Wah Wu
def a(n): return a069010(a087207(n)) # Indranil Ghosh, Jun 06 2017

a(n) = A069010(A087207(n))

A356233 Number of integer factorizations of n into gapless numbers (A066311).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 2, 5, 1, 4, 1, 2, 1, 1, 1, 7, 2, 1, 3, 2, 1, 4, 1, 7, 1, 1, 2, 9, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 12, 2, 2, 1, 2, 1, 7, 1, 3, 1, 1, 1, 8, 1, 1, 2, 11, 1, 2, 1, 2, 1, 2, 1, 16, 1, 1, 4, 2, 2, 2, 1, 5, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 28 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. We define a number to be gapless (listed by A066311) iff its prime indices cover an interval of positive integers.

			The counted factorizations of n = 2, 4, 8, 12, 24, 36, 48:
  (2)  (4)    (8)      (12)     (24)       (36)       (48)
       (2*2)  (2*4)    (2*6)    (3*8)      (4*9)      (6*8)
              (2*2*2)  (3*4)    (4*6)      (6*6)      (2*24)
                       (2*2*3)  (2*12)     (2*18)     (3*16)
                                (2*2*6)    (3*12)     (4*12)
                                (2*3*4)    (2*2*9)    (2*3*8)
                                (2*2*2*3)  (2*3*6)    (2*4*6)
                                           (3*3*4)    (3*4*4)
                                           (2*2*3*3)  (2*2*12)
                                                      (2*2*2*6)
                                                      (2*2*3*4)
                                                      (2*2*2*2*3)

The shortest of these factorizations is listed at A356234, length A287170.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356226 lists the lengths of maximal gapless submultisets of prime indices:
- length: A287170
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232
Cf. A001222, A060680-A060683, A073491-A073495, A193829, A328195, A328335-A328458.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sqq[n_]:=Max@@Differences[primeMS[n]]<=1;
Table[Length[Select[facs[n],And@@sqq/@#&]],{n,100}]

A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 10 2022

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.

			Triangle  begins: {}, {1}, {1}, {2}, {1}, {2}, {1}, {3}, {2}, {1,1}, {1}, {3}, {1}, {1,1}, {2}, {4}, {1}, {3}, {1}, {2,1}, ... For example, the prime indices of 20 are {1,1,3}, which separates into maximal gapless submultisets {{1,1},{3}}, so row 20 is (2,1).
The prime indices of 18564 are {1,1,2,4,6,7}, which separates into {1,1,2}, {4}, {6,7}, so row 18564 is (3,1,2). This corresponds to the factorization 18564 = 12 * 7 * 221.

Row sums are A001222.
Singleton row positions are A073491, complement A073492.
Length-2,3,4 row positions are A073493-A073495.
Row lengths are A287170, firsts A066205.
Row minima are A356227.
Row maxima are A356228.
Bisected run-lengths are A356229.
Standard composition numbers of rows are A356230.
Heinz numbers of rows are A356231.
Positions of first appearances are A356232.
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A055874, A060680-A060683, A137921, A193829, A286470, A328166.

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

A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 3, 1, 4, 1, 3, 2, 8, 1, 4, 1, 5, 3, 3, 1, 8, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 2, 8, 1, 3, 3, 9, 1, 5, 1, 5, 4, 3, 1, 16, 2, 6, 3, 5, 1, 8, 3, 9, 3, 3, 1, 8, 1, 3, 5, 32, 3, 5, 1, 5, 3, 6, 1, 16, 1, 3, 4, 5, 2, 5, 1, 17, 8, 3, 1, 9, 3

Offset: 1

Gus Wiseman, Aug 16 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.
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), which is the 38th composition in standard order, so a(18564) = 38.

Numbers grouped by number of gaps in prime indices are A073491-A073495.
These are the standard composition numbers of rows of A356226.
Using Heinz numbers instead of standard compositions gives A356231.
Positions of first appearances are A356603, sorted A356232.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A066099 lists compositions in standard order.
A132747 counts non-isolated divisors, complement A132881.
A333627 represents the run-lengths of standard compositions.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000120, A001222, A060680-A060683, A066205, A286470, A287170, A328166, A356227, A356228, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

A000120(a(n)) = A287170(n).
A333766(a(n)) = A356228(n).
A333768(a(n)) = A356227(n).

A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 160, 200, 220, 250, 256, 320, 400, 440, 500, 512, 550, 640, 800, 880, 1000, 1024, 1100, 1210, 1250, 1280, 1600, 1760, 1870, 2000, 2048, 2200, 2420, 2500, 2560, 2750, 3200, 3520, 3740, 4000, 4096, 4400

Offset: 1

Gus Wiseman, Aug 20 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.

Also positions of first appearances of rows in A356226.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     10: {1,3}
     16: {1,1,1,1}
     20: {1,1,3}
     32: {1,1,1,1,1}
     40: {1,1,1,3}
     50: {1,3,3}
     64: {1,1,1,1,1,1}
     80: {1,1,1,1,3}
    100: {1,1,3,3}
    110: {1,3,5}
    128: {1,1,1,1,1,1,1}
    160: {1,1,1,1,1,3}
    200: {1,1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    256: {1,1,1,1,1,1,1,1}
    320: {1,1,1,1,1,1,3}
    400: {1,1,1,1,3,3}

The partitions with these Heinz numbers are counted by A053251.
This is the odd restriction of A055932.
A subset of A066208 (numbers with all odd prime indices).
This is the sorted version of A356603.
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
- positions of first appearances: A356232 (this sequence)
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A001222, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[1000],normQ[(primeMS[#]+1)/2]&]

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

Original entry on oeis.org

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).

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]]}]

cp's OEIS Frontend

A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

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A356233 Number of integer factorizations of n into gapless numbers (A066311).

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A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

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A356230 The a(n)-th composition in standard order is the sequence of lengths of maximal gapless submultisets of the prime indices of n.

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A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

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

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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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