A164894 -id:A164894 - cp's OEIS frontend

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Mar 23 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. See also A000120, A059893, A070939, A114994, A225620.

			The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positive positions of first appearances are A164894.
The version for partitions is A257990.
The strong opposite version is A352514, counted by A352521 (first A219282).
The opposite version is A352515, counted by A352522 (first column A238874).
The strong version is A352516, counted by A352524 (first column A008930).
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 is the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
Table[pdw[stc[n]],{n,0,30}]

A124764 Number of non-falls (levels or rises) for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is one fewer than the number of maximal strictly decreasing runs in this composition. Alternatively, a(n) is the number of weak ascents in the same composition. For example, the strictly decreasing runs of the 1234567th composition are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so a(1234567) = 7 - 1 = 6. The 6 weak ascents together with the strict descents are: 3 > 2 > 1 <= 2 <= 2 > 1 <= 2 <= 5 > 1 <= 1 <= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>1<=1, so a(11) = 1.
The table starts:
  0
  0
  0 1
  0 0 1 2
  0 0 1 1 1 1 2 3
  0 0 0 1 1 1 1 2 1 1 2 2 2 2 3 4
  0 0 0 1 1 0 1 2 1 1 2 2 1 1 2 3 1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5

Cf. A066099, A124760, A124761, A124762, A124763, A011782 (row lengths), A045883 (row sums), A233249, A238343.
Compositions of n with k weak ascents are A333213.
Positions of zeros are A333256.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793 (triangle).
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Reversed initial intervals A164894.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],LessEqual@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(n) = A124760(n) + A124762(n)
For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1=b(i+1)} 1.
For n > 0, a(n) = A124769(n) - 1. - Gus Wiseman, Apr 08 2020

A333379 Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 6, 7, 14, 15, 26, 30, 31, 52, 58, 62, 63, 106, 116, 122, 126, 127, 212, 234, 244, 250, 254, 255, 420, 426, 468, 490, 500, 506, 510, 511, 840, 852, 932, 938, 980, 1002, 1012, 1018, 1022, 1023, 1700, 1706, 1864, 1876, 1956, 1962, 2004, 2026, 2036, 2042

Offset: 1

Gus Wiseman, Mar 21 2020

nonn

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.

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              212: (1,2,2,3)
    3: (1,1)            234: (1,1,2,2,2)
    6: (1,2)            244: (1,1,1,2,3)
    7: (1,1,1)          250: (1,1,1,1,2,2)
   14: (1,1,2)          254: (1,1,1,1,1,1,2)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   26: (1,2,2)          420: (1,2,3,3)
   30: (1,1,1,2)        426: (1,2,2,2,2)
   31: (1,1,1,1,1)      468: (1,1,2,2,3)
   52: (1,2,3)          490: (1,1,1,2,2,2)
   58: (1,1,2,2)        500: (1,1,1,1,2,3)
   62: (1,1,1,1,2)      506: (1,1,1,1,1,2,2)
   63: (1,1,1,1,1,1)    510: (1,1,1,1,1,1,1,2)
  106: (1,2,2,2)        511: (1,1,1,1,1,1,1,1,1)
  116: (1,1,2,3)        840: (1,2,3,4)
  122: (1,1,1,2,2)      852: (1,2,2,2,3)
  126: (1,1,1,1,1,2)    932: (1,1,2,3,3)

Sequences covering an initial interval are counted by A000670.
Compositions in standard order are A066099.
Weakly increasing runs are counted by A124766.
Removing the covering condition gives A225620.
Removing the ordering condition gives A333217.
The strictly increasing case is A164894.
The strictly decreasing version is A246534.
The unequal version is A333218.
The weakly decreasing version is A333380.
Cf. A000120, A000225, A029931, A048793, A070939, A228351, A233564, A272919.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&LessEqual@@stc[#]&]

Intersection of A333217 and A225620.

A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

Original entry on oeis.org

1, 6, 7, 52, 13, 222, 15, 840, 121, 858, 27, 28268, 29, 894, 991, 26896, 49, 113970, 51, 215892, 2037, 3446, 55, 14471576, 441, 3514, 3899, 217052, 61, 14538238, 63, 1721376, 7905, 13410, 7139, 926213284, 101, 13542, 8039, 221009192

Offset: 1

Omar E. Pol, Nov 22 2010

a(n) is A182621(n), interpreted as a binary number, written in base 10. The first repeated element is 991, from 15 and 479.

Except for 1, no power of 2 can occur in this sequence, an obvious consequence of the fact that a(n) has to be the sum of at least two distinct powers of 2 for all n > 1. - Alonso del Arte, Nov 13 2013

			The divisors of 10 are 1, 2, 5, 10. Then 1, 2, 5, 10 written in base 2 are 1, 10, 101, 1010. The concatenation of 1, 10, 101, 1010 is 1101011010. Then a(10) = 858 because the binary number 1101011010 written in base 10 is 858.

Indranil Ghosh, Table of n, a(n) for n = 1..50000

Cf. A027750, A007088, A182620, A182621, A182623, A182624, A182627, A182632.

concatBits[n_] := FromDigits[Join @@ (IntegerDigits[#, 2]& /@ Divisors[n]), 2]; concatBits /@ Range[40](* Giovanni Resta, Nov 23 2010 *)

a(n) = {my(cbd = []); fordiv(n, d, cbd = concat(cbd, binary(d));); fromdigits(cbd, 2);} \\ Michel Marcus, Jan 28 2017

def A182622(n):
    s=""
    for i in range(1,n+1):
        if n%i==0:
            s+=bin(i)[2:]
    return int(s,2) # Indranil Ghosh, Jan 28 2017

a(p) = 2^(floor(log_2(p)) + 1) + p for p prime. Also, a(p + k) > a(p) for all k > 0. Furthermore, for all primes p > 3, a(p) < a(p - 1).

a(2^(m - 1)) = sum(k = 0 .. m - 1, 2^((m^2 + m)/2 - (k^2 + k)/2 - 1)) = A164894(m). - Alonso del Arte, Nov 13 2013

More terms from Giovanni Resta, Nov 23 2010

A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 21, 23, 31, 37, 43, 47, 63, 75, 85, 87, 95, 127, 149, 151, 171, 175, 191, 255, 293, 299, 303, 341, 343, 351, 383, 511, 549, 587, 597, 599, 607, 683, 687, 703, 767, 1023, 1099, 1173, 1175, 1195, 1199, 1215, 1365, 1367, 1375, 1407, 1535

Offset: 1

Gus Wiseman, Mar 21 2020

nonn

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.

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              149: (3,2,2,1)
    3: (1,1)            151: (3,2,1,1,1)
    5: (2,1)            171: (2,2,2,1,1)
    7: (1,1,1)          175: (2,2,1,1,1,1)
   11: (2,1,1)          191: (2,1,1,1,1,1,1)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   21: (2,2,1)          293: (3,3,2,1)
   23: (2,1,1,1)        299: (3,2,2,1,1)
   31: (1,1,1,1,1)      303: (3,2,1,1,1,1)
   37: (3,2,1)          341: (2,2,2,2,1)
   43: (2,2,1,1)        343: (2,2,2,1,1,1)
   47: (2,1,1,1,1)      351: (2,2,1,1,1,1,1)
   63: (1,1,1,1,1,1)    383: (2,1,1,1,1,1,1,1)
   75: (3,2,1,1)        511: (1,1,1,1,1,1,1,1,1)
   85: (2,2,2,1)        549: (4,3,2,1)
   87: (2,2,1,1,1)      587: (3,3,2,1,1)
   95: (2,1,1,1,1,1)    597: (3,2,2,2,1)

Sequences covering an initial interval are counted by A000670.
Compositions in standard order are A066099.
Weakly decreasing runs are counted by A124765.
Removing the covering condition gives A114994.
Removing the ordering condition gives A333217.
The strictly decreasing case is A246534.
The unequal version is A333218.
The weakly increasing version is A333379.
Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]

Intersection of A333217 and A114994.

A331579 Position of first appearance of n in A124758 (products of compositions in standard order).

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 64, 34, 36, 66, 1024, 68, 4096, 258, 132, 136, 65536, 146, 262144, 264, 516, 4098

Offset: 1

Gus Wiseman, Mar 20 2020

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.

			The list of terms together with the corresponding compositions begins:
       1: (1)
       2: (2)
       4: (3)
       8: (4)
      16: (5)
      18: (3,2)
      64: (7)
      34: (4,2)
      36: (3,3)
      66: (5,2)
    1024: (11)
      68: (4,3)
    4096: (13)
     258: (7,2)
     132: (5,3)
     136: (4,4)
   65536: (17)
     146: (3,3,2)
  262144: (19)
     264: (5,4)

The product of prime indices is A003963.
The sum of binary indices is A029931.
The sum of prime indices is A056239.
Sums of compositions in standard order are A070939.
The product of binary indices is A096111.
All terms belong to A114994.
Products of compositions in standard order are A124758.
Cf. A000120, A048793, A066099, A164894, A233249, A233564, A272919, A326674, A333217, A333219, A333220, A333223.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
q=Table[Times@@stc[n],{n,1000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A104462 Convert the binary strings in A101305 to decimal.

Original entry on oeis.org

0, 2, 20, 328, 10512, 672800, 86118464, 22046326912, 11287719379200, 11558624644301312, 23672063271529088000, 96960771160183144450048, 794302637344220319334797312, 13013854410247705711981319168000, 426437981314996820770203866497040384

Offset: 0

Jorge Coveiro, Apr 23 2005

The a(n)-th composition in standard order is (2,3,..,n+1), where 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. Moreover, the binary indices of a(n) are row n of A193973. Including 1 gives A164894, reverse A246534. - Gus Wiseman, Jun 28 2022

			From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their standard compositions begin:
      0: ()
      2: (2)
     20: (2,3)
    328: (2,3,4)
  10512: (2,3,4,5)
(End)

Michael S. Branicky, Table of n, a(n) for n = 0..80

Cf. A101305.
A version for prime indices is A070826.
Cf. A000120, A002110, A029931, A066099, A070939, A164894, A193973, A233564, A246534, A272919, A333218, A333255.

convert(10,decimal,binary); convert(10100,decimal,binary); convert(101001000,decimal,binary); convert(10100100010000,decimal,binary); convert(10100100010000100000,decimal,binary);

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Range[2,n]],{n,8}] (* Gus Wiseman, Jun 28 2022 *)

def a(n): return 0 if n==0 else int("".join("1"+"0"*(i+1) for i in range(n)), 2)
print([a(n) for n in range(15)]) # Michael S. Branicky, Jun 28 2022

a(14) and beyond from Michael S. Branicky, Jun 28 2022

A357428 Numbers whose digit representation in base 2 is equal to the digit representation in base 2 of the initial terms of their sets of divisors in increasing order.

Original entry on oeis.org

1, 6, 52, 63, 222, 2037, 6776, 26896, 124641, 220336192, 222066488

Offset: 1

Michel Marcus, Sep 28 2022

a(1), a(2), a(3), a(8) and a(10) belong to A164894; A164894(13) = 2032242676629600594233921536, A164894(19) = 1288086824419468350412109535086131006200927555108489920512 and A164894(29) are also terms. - Rémy Sigrist, Sep 28 2022

			In base 2, 6 is 110 and its first divisors are 1 and 2, that is, 1 and 10.

Cf. A164894, A175252 (base 10), A357429 (base 3).

isok(k) = my(s=[]); fordiv(k, d, s=concat(s, binary(d)); if (fromdigits(s, 2)==k, return(1)); if (fromdigits(s,2)> k, return(0)));

from sympy import divisors
def ok(n):
    target, s = bin(n)[2:], ""
    if target[0] != "1": return False
    for d in divisors(n):
        s += bin(d)[2:]
        if len(s) >= len(target): return s == target
        elif not target.startswith(s): return False
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 01 2022

a(10)-a(11) from Rémy Sigrist, Sep 28 2022

A309079 For any n > 0: consider the strictly increasing finite sequences of integers whose concatenation of terms, in binary and without leading zeros, equals that of n; a(n) is the minimal sum of the terms of such a finite sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 8, 9, 10, 5, 5, 6, 7, 8, 16, 17, 18, 19, 6, 7, 8, 9, 9, 10, 11, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 9, 10, 11, 10, 11, 12, 13, 14, 15, 11, 12, 17, 18, 19, 20, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 64, 65, 66, 67, 68, 69, 70, 71, 12

Offset: 1

Rémy Sigrist, Jul 11 2019

Any integer appear in the sequence:
- for any m > 0 with binary expansion Sum_{k >= 0} b_k * 2^k,
- let n = (Sum_{k >= 0} b_k * 2^Sum_{j >= k} ((1+j) * b_j))/2,
- then a(n) = m,
- for example (in binary): a("1101000") = "1" + "10" + "1000" = "1011".

			The first terms, alongside the corresponding finite sequences, are:
  n   a(n)  bin(n)  bin(seq)
  --  ----  ------  --------
   1     1       1  (1)
   2     2      10  (10)
   3     3      11  (11)
   4     4     100  (100)
   5     5     101  (101)
   6     3     110  (1,10)
   7     4     111  (1,11)
   8     8    1000  (1000)
   9     9    1001  (1001)
  10    10    1010  (1010)
  11     5    1011  (10,11)
  12     5    1100  (1,100)
  13     6    1101  (1,101)
  14     7    1110  (1,110)
  15     8    1111  (1,111)
  16    16   10000  (10000)
  17    17   10001  (10001)
  18    18   10010  (10010)
  19    19   10011  (10011)
  20     6   10100  (10,100)
  21     7   10101  (10,101)

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A309079

Cf. A000225, A143789, A164894.

PARI
```
See Links section.
```

a(n) <= n with equality iff n is a power of two or the binary concatenation of 2^k and m for some k >= 0 and m <= 2^k.
a(2*n) <= 2*a(n).
a(2*n + 1) <= 2*a(n) + 1.
a(A164894(k)) = A000225(k) for any k > 0.

A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

Original entry on oeis.org

37, 52, 549, 550, 556, 564, 581, 600, 616, 649, 657, 712, 786, 802, 836, 840, 16933, 16934, 16937, 16940, 16946, 16948, 16965, 16977, 16984, 16994, 17000, 17033, 17041, 17092, 17096, 17170, 17186, 17220, 17224, 17445, 17446, 17452, 17460, 17541, 17569, 17584

Offset: 1

Gus Wiseman, Jan 13 2022

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.

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 corresponding permutations begin:
     37: (3,2,1)
     52: (1,2,3)
    549: (4,3,2,1)
    550: (4,3,1,2)
    556: (4,2,1,3)
    564: (4,1,2,3)
    581: (3,4,2,1)
    600: (3,2,1,4)
    616: (3,1,2,4)
    649: (2,4,3,1)
    657: (2,3,4,1)
    712: (2,1,3,4)
    786: (1,4,3,2)
    802: (1,3,4,2)
    836: (1,2,4,3)
    840: (1,2,3,4)
  16933: (5,4,3,2,1)

This is the non-alternating case of A333218.
This is the restriction of A345168 to permutations, complement A345167.
These partitions are counted by A348615, complement A001250.
A003242 counts anti-run compositions, patterns A005649.
A025047 counts alternating compositions, directed A025048/A025049.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns, complement A350252.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994, strict A333256.
- Weakly increasing compositions (multisets) are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489, complement A348612.
Cf. A008965, A059893, A164894, A246534, A333217, A344605, A345162, A350251, A345163, A345171, A345172, A348613.

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[#]])&]

cp's OEIS Frontend

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

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A124764 Number of non-falls (levels or rises) for compositions in standard order.

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A333379 Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.

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A182622 a(n) is the number whose binary representation is the concatenation of the divisors of n written in base 2.

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PARI

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A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

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A331579 Position of first appearance of n in A124758 (products of compositions in standard order).

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A104462 Convert the binary strings in A101305 to decimal.

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Maple

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A357428 Numbers whose digit representation in base 2 is equal to the digit representation in base 2 of the initial terms of their sets of divisors in increasing order.

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