A073493 -id:A073493 - cp's OEIS frontend

A073492 Numbers having at least one prime gap in their factorization.

10, 14, 20, 21, 22, 26, 28, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073490(a(n)) > 0.

A137794(a(n))=0, complement of A073491. - Reinhard Zumkeller, Feb 11 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A073493, A073494, A073495, A073491, A073486.

a073492 n = a073492_list !! (n-1)
a073492_list = filter ((> 0) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[126],Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] > 0 &] (* Jayanta Basu, Jul 01 2013 *)

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

Original entry on oeis.org

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))

A107428 Number of gap-free compositions of n.

Original entry on oeis.org

1, 2, 4, 6, 11, 21, 39, 71, 141, 276, 542, 1070, 2110, 4189, 8351, 16618, 33134, 66129, 131937, 263483, 526453, 1051984, 2102582, 4203177, 8403116, 16800894, 33593742, 67174863, 134328816, 268624026, 537192064, 1074288649, 2148414285, 4296543181, 8592585289

Offset: 1

N. J. A. Sloane, May 26 2005

nonn

A gap-free composition contains all the parts between its smallest and largest part. a(5)=11 because we have: 5, 3+2, 2+3, 2+2+1, 2+1+2, 1+2+2, 2+1+1+1, 1+2+1+1, 1+1+2+1, 1+1+1+2, 1+1+1+1+1. - Geoffrey Critzer, Apr 13 2014

			From _Gus Wiseman_, Oct 04 2022: (Start)
The a(0) = 1 through a(5) = 11 gap-free compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (22)    (23)
                 (21)   (112)   (32)
                 (111)  (121)   (122)
                        (211)   (212)
                        (1111)  (221)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 42..1000
P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.

The unordered version (partitions) is A034296, ranked by A073491.
The initial case is A107429, unordered A000009, ranked by A333217.
The unordered complement is counted by A239955, ranked by A073492.
These compositions are ranked by A356841.
The complement is counted by A356846, ranked by A356842
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A055932, A073493, A137921, A251729, A356224, A356843, A356845.

b:= proc(n, i, t) option remember; `if`(n=0, t!,
      `if`(i<1 or n add(b(n, i, 0), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 14 2014

Table[Length[Select[Level[Map[Permutations,IntegerPartitions[n]],{2}],Length[Union[#]]==Max[#]-Min[#]+1&]],{n,1,20}] (* Geoffrey Critzer, Apr 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) ~ 2^(n-2). - Alois P. Heinz, Dec 07 2014

G.f.: Sum_{j>0} Sum_{k>=j} C({j..k},x) where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) is the g.f. for compositions such that the set of parts equals {s} with C({},x) = 1. - John Tyler Rascoe, Jun 01 2024

More terms from Vladeta Jovovic, May 26 2005

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}]

A073490 Number of prime gaps in factorization of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=kA137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

			84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;
110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.

R. Zumkeller, Table of n, a(n) for n = 1..50000

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

a073490 1 = 0
a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips
   where ips = map a049084 $ a027748_row n
-- Reinhard Zumkeller, Jul 04 2012

A073490 := proc(n)
    local a,plist ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    a := 0 ;
    for i from 2 to nops(plist) do
        if op(i,plist) <> nextprime(op(i-1,plist)) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A073490(n),n=1..110) ; # R. J. Mathar, Oct 27 2019

gaps[n_Integer/;n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 120}] (* Wouter Meeussen, Oct 30 2004 *)
pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

from sympy import primefactors, nextprime
def a(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))
print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

a(n) = A073484(A007947(n)).
a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n) = 1.
a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

More terms from Franklin T. Adams-Watters, May 19 2006

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).

A073495 Numbers having exactly three prime gaps in their factorization.

Original entry on oeis.org

1870, 2090, 2470, 2530, 2990, 3190, 3410, 3458, 3740, 3770, 3910, 4030, 4070, 4180, 4186, 4510, 4730, 4810, 4930, 4940, 5060, 5170, 5187, 5270, 5278, 5330, 5474, 5510, 5590, 5642, 5830, 5890, 5980, 6110, 6279, 6290, 6380, 6490, 6710, 6734, 6820, 6890

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			1870 is a term, as 1870 = 2*5*11*17 with three gaps: between 2 and 5, between 5 and 11 and between 11 and 17.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A073490, A073492, A073493, A073494, A073489.

a073495 n = a073495_list !! (n-1)
a073495_list = filter ((== 3) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True] == 3; Select[Range[7000], q] (* Amiram Eldar, Apr 10 2021*)

A073490(a(n)) = 3.

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]]}]

A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 64, 68, 72, 74, 75, 77, 78, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101

Offset: 1

Gus Wiseman, Aug 31 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 terms and their corresponding standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)

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

See link for sequences related to standard compositions.
An unordered version is A073491, complement A073492.
These compositions are counted by A107428.
The complement is A356842.
The non-initial case is A356843, unordered A356845.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224/A356225.

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],nogapQ[stc[#]]&]

cp's OEIS Frontend

A073492 Numbers having at least one prime gap in their factorization.

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

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A107428 Number of gap-free compositions of n.

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

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A073490 Number of prime gaps in factorization 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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