A333766 -id:A333766 - cp's OEIS frontend

A003754 Numbers with no adjacent 0's in binary expansion.

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
The k-th composition in standard order (row k of 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. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Aug 14 2003

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k > 0; A087116(n) = 0; A023416(n) = 0.

The k-th composition in standard order (row k of 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. Then a(k) is the maximum part of this composition, minus one. The maximum part is A333766(k). - Gus Wiseman, Apr 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A023416, A007088, A007814.
Cf. A025480, A038374, A090046, A090047, A090048, A090049, A090050.
Positions of zeros are A000225.
Positions of terms <= 1 are A003754.
Positions of terms > 0 are A062289.
Positions of first appearances are A131577.
The version for prime indices is A252735.
The proper maximum is A333766.
The version for minimum is A333767.
Maximum prime index is A061395.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A061395, A087117, A228351, A328594, A333217, A333218, A333219, A333767, A333768.

import Data.List (unfoldr, group)
a087117 0       = 1
a087117 n
  | null $ zs n = 0
  | otherwise   = maximum $ map length $ zs n where
  zs = filter ((== 0) . head) . group .
       unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, May 01 2012

A087117 := proc(n)
    local d,l,zlen ;
    if n = 0 then
        return 1 ;
    end if;
    d := convert(n,base,2) ;
    for l from nops(d)-1 to 0 by -1 do
        zlen := [seq(0,i=1..l)] ;
        if verify(zlen,d,'sublist') then
            return l ;
        end if;
    end do:
    return 0 ;
end proc; # R. J. Mathar, Nov 05 2012

nz[n_]:=Max[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]]; Array[nz,110,0]/.-\[Infinity]->0 (* Harvey P. Dale, Sep 05 2017 *)

h(n)=if(n<2, return(0)); my(k=valuation(n,2)); if(k, max(h(n>>k), k), n++; n>>=valuation(n,2); h(n-1))
a(n)=if(n,h(n),1) \\ Charles R Greathouse IV, Apr 06 2022

a(n) = max(A007814(n), a(A025480(n-1))) for n >= 2. - Robert Israel, Feb 19 2017
a(2n+1) = a(n) (n>=1); indeed, the binary form of 2n+1 consists of the binary form of n with an additional 1 at the end - Emeric Deutsch, Aug 18 2017
For n > 0, a(n) = A333766(n) - 1. - Gus Wiseman, Apr 09 2020

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).

A333768 Minimum part of the n-th composition in standard order. a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 0

Gus Wiseman, Apr 06 2020

nonn

One plus the shortest run of 0's after a 1 in the binary expansion of n > 0.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 148th composition in standard order is (3,2,3), so a(148) = 2.

Positions of first appearances (ignoring index 0) are A000079.
Positions of terms > 1 are A022340.
The version for prime indices is A055396.
The  maximum part is given by A333766.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Min@@stc[n]],{n,0,100}]

For n > 0, a(n) = A333767(n) + 1.

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)).

A349051 Numbers k such that the k-th composition in standard order is an alternating permutation of {1..k} for some k.

Original entry on oeis.org

0, 1, 5, 6, 38, 41, 44, 50, 553, 562, 582, 593, 610, 652, 664, 708, 788, 808, 16966, 17036, 17048, 17172, 17192, 17449, 17458, 17542, 17676, 17712, 17940, 18000, 18513, 18530, 18593, 18626, 18968, 18992, 19496, 19536, 20625, 20676, 20769, 20868, 21256, 22600

Offset: 1

Gus Wiseman, Nov 08 2021

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 sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence together with the corresponding compositions begins:
        0: ()
        1: (1)
        5: (2,1)
        6: (1,2)
       38: (3,1,2)
       41: (2,3,1)
       44: (2,1,3)
       50: (1,3,2)
      553: (4,2,3,1)
      562: (4,1,3,2)
      582: (3,4,1,2)
      593: (3,2,4,1)
      610: (3,1,4,2)
      652: (2,4,1,3)
      664: (2,3,1,4)
      708: (2,1,4,3)
      788: (1,4,2,3)
      808: (1,3,2,4)
    16966: (5,3,4,1,2)
    17036: (5,2,4,1,3)

Wikipedia, Alternating permutation

These permutations are counted by A001250, complement A348615.
Compositions of this type are counted by A025047, complement A345192.
Subset of A333218, which ranks permutations of initial intervals.
Subset of A345167, which ranks alternating compositions, complement A345168.
A003242 counts Carlitz (anti-run) compositions.
A345163 counts normal partitions with an alternating permutation.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with an alternating permutation.
Compositions in standard order are the rows of A066099:
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- GCD and LCM are given by A326674 and A333226.
- Maximal runs and anti-runs are counted by A124767 and A333381.
- Heinz number is given by A333219.
- Runs-resistance is given by A333628.
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz (anti-run) compositions are ranked by A333489.
Cf. A025048, A025049, A344604, A344615, A344653, A344742, A348377, A348379, A345165.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,1000],Sort[stc[#]]==Range[Length[stc[#]]]&&wigQ[stc[#]]&]

Equals A333218 (permutation) /\ A345167 (alternating).

A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 3, 2, 2, 1, 5, 4, 3, 3, 2, 2, 1, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 7, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 1, 8, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 9, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1

Offset: 1

Gus Wiseman, May 08 2020

The first partition ranked by A080577 is (); there is no zeroth partition.

			The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.
  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (2,1,1,1)    (6,1)          (8)
  (3)        (1,1,1,1,1)  (5,2)          (7,1)
  (2,1)      (6)          (5,1,1)        (6,2)
  (1,1,1)    (5,1)        (4,3)          (6,1,1)
  (4)        (4,2)        (4,2,1)        (5,3)
  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
Triangle begins:
  0
  1
  2 1
  3 2 1
  4 3 2 2 1
  5 4 3 3 2 2 1
  6 5 4 4 3 3 3 2 2 2 1
  7 6 5 5 4 4 4 3 3 3 3 2 2 2 1
  8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
The version for compositions is A065120 or A333766.
Reverse-lexicographically ordered partitions are A080577.
Distinct parts of these partitions are counted by A115623.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Lengths of these partitions are A238966.
Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Prepend[First/@Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,8}],0]

a(n) = A061395(A129129(n - 1)).

A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 26 2022

nonn

			Triangle begins:
   0:
   1: 1
   2: 2
   3: 1 1
   4: 1 1
   5: 2 1
   6: 1 2
   7: 1 1 1
   8: 3
   9: 1 1 1
  10: 2 2
  11: 2 1 1
  12: 1 1 1
  13: 1 2 1
  14: 1 1 2
  15: 1 1 1 1

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

See link for sequences related to standard compositions.
Row n is the A357134(n)-th composition in standard order.
The version for Heinz numbers of partitions is A357139, cf. A003963.
Row sums are A357186, differences A357187.
Cf. A000120, A001511, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Join@@Table[Join@@stc/@stc[n],{n,0,30}]

Row n is the A357134(n)-th composition in standard order.

A357134 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then set a(n) to be the index (in standard order) of the concatenation.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 6, 7, 4, 7, 10, 11, 7, 13, 14, 15, 5, 9, 14, 15, 11, 21, 22, 23, 12, 15, 26, 27, 15, 29, 30, 31, 6, 11, 18, 19, 15, 29, 30, 31, 20, 23, 42, 43, 23, 45, 46, 47, 13, 25, 30, 31, 27, 53, 54, 55, 28, 31, 58, 59, 31, 61, 62, 63, 7, 13, 22, 23, 19

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

			The terms together with their corresponding standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   3: (1,1)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   4: (3)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
   7: (1,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)

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

See link for sequences related to standard compositions.
The version for Heinz numbers of partitions is A003963.
The vertex-degrees are A048896.
The a(n)-th composition in standard order is row n of A357135.
Cf. A000120, A001511, A029931, A058891, A070939, A096111, A329395, A333766, A335404, A357137, A357180.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Join@@stc/@stc[n]],{n,0,30}]

For n > 0, the value n appears A048896(n - 1) times.

Row a(n) of A066099 = row n of A357135.

A357137 Maximal run-length of the n-th composition in standard order; a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 18 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

			Composition 92 in standard order is (2,1,1,3), so a(92) = 2.

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

See link for more sequences related to standard compositions.
The version for Heinz numbers of partitions is A051903, for parts A061395.
For parts instead of run-lengths we have A333766, minimal A333768.
The opposite (minimal) version is A357138.
For first instead of maximal we have A357180, last A357181.
Cf. A000120, A001511, A003754, A029931, A051904, A055396, A056239, A070939, A286470, A356844, A357136.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Max[Length/@Split[stc[n]]]],{n,0,100}]

cp's OEIS Frontend

A003754 Numbers with no adjacent 0's in binary expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A333768 Minimum part of the n-th composition in standard order. a(0) = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A349051 Numbers k such that the k-th composition in standard order is an alternating permutation of {1..k} for some k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs