A029931 -id:A029931 - cp's OEIS frontend

A326704 BII-numbers of antichains of nonempty sets.

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524

Offset: 1

Gus Wiseman, Jul 21 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.

			The sequence of all antichains of nonempty sets together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..7580
John Tyler Rascoe, Python Program.

Antichains of sets are counted by A000372.
Antichains of nonempty sets are counted by A014466.
MM-numbers of antichains of multisets are A316476.
BII-numbers of chains of nonempty sets are A326703.
Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[bpe/@bpe[#],SubsetQ]&]

Python
```
# see linked program
```

A326749 BII-numbers of connected set-systems.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, Jul 23 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all connected set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Positions of 0's and 1's in A326753.
Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A323818, A326031, A326702.
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[0,100],Length[csm[bpe/@bpe[#]]]<=1&]

from itertools import count, islice
from sympy.utilities.iterables import connected_components
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    yield 0
    for n in count(1):
        a, E = [bin_i(k) for k in bin_i(n)], []
        m = len(a)
        for i in range(m):
            for j in a[i]:
                for k in range(m):
                    if j in a[k]:
                        E.append((i, k))
        for v in connected_components((list(range(m)), E)):
            if len(v) == m:
                yield n
A326749_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jul 25 2024

A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 32, 33, 34, 37, 64, 65, 66, 68, 69, 128, 129, 130, 132, 133, 137, 256, 257, 258, 260, 261, 264, 265, 274, 512, 513, 514, 516, 517, 520, 521, 529, 530, 549, 1024, 1025, 1026, 1028, 1029, 1032, 1033, 1040, 1041, 1042, 1058, 1061

Offset: 1

Gus Wiseman, Mar 20 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 positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         517: (7,2,1)
     2: (2)         129: (7,1)       520: (6,4)
     4: (3)         130: (6,2)       521: (6,3,1)
     5: (2,1)       132: (5,3)       529: (5,4,1)
     8: (4)         133: (5,2,1)     530: (5,3,2)
     9: (3,1)       137: (4,3,1)     549: (4,3,2,1)
    16: (5)         256: (9)        1024: (11)
    17: (4,1)       257: (8,1)      1025: (10,1)
    18: (3,2)       258: (7,2)      1026: (9,2)
    32: (6)         260: (6,3)      1028: (8,3)
    33: (5,1)       261: (6,2,1)    1029: (8,2,1)
    34: (4,2)       264: (5,4)      1032: (7,4)
    37: (3,2,1)     265: (5,3,1)    1033: (7,3,1)
    64: (7)         274: (4,3,2)    1040: (6,5)
    65: (6,1)       512: (10)       1041: (6,4,1)
    66: (5,2)       513: (9,1)      1042: (6,3,2)
    68: (4,3)       514: (8,2)      1058: (5,4,2)
    69: (4,2,1)     516: (7,3)      1061: (5,3,2,1)

Strictly increasing runs are counted by A124768.
The normal case is A246534.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly increasing version is A333255.
Cf. A000120, A029931, A048793, A066099, A070939, A124769, A228351, A233249, A333217, A333256, A333380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Greater@@stc[#]&]

A291166 Connected Haar graph numbers.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Eric W. Weisstein, Aug 19 2017

nonn

Complement of A291165.

These appear to be numbers whose positions of 1's in their reversed binary expansion are relatively prime. If so, this sequence lists all positions of 1's in A326674. Numbers whose positions of 1's in their reversed binary expansion are pairwise coprime (as opposed to relatively prime) are A326675. - Gus Wiseman, Jul 19 2019

Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A000120, A029931, A048793, A059519, A070939, A289508, A291165, A295235, A319826, A326031, A326669, A326672, A326673.

A333255 Numbers k such that the k-th composition in standard order is strictly increasing.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 52, 64, 72, 80, 96, 104, 128, 144, 160, 192, 200, 208, 256, 272, 288, 320, 328, 384, 400, 416, 512, 544, 576, 640, 656, 768, 784, 800, 832, 840, 1024, 1056, 1088, 1152, 1280, 1296, 1312, 1536, 1568, 1600, 1664, 1680

Offset: 1

Gus Wiseman, Mar 20 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 positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         656: (2,3,5)
     2: (2)         144: (3,5)       768: (1,9)
     4: (3)         160: (2,6)       784: (1,4,5)
     6: (1,2)       192: (1,7)       800: (1,3,6)
     8: (4)         200: (1,3,4)     832: (1,2,7)
    12: (1,3)       208: (1,2,5)     840: (1,2,3,4)
    16: (5)         256: (9)        1024: (11)
    20: (2,3)       272: (4,5)      1056: (5,6)
    24: (1,4)       288: (3,6)      1088: (4,7)
    32: (6)         320: (2,7)      1152: (3,8)
    40: (2,4)       328: (2,3,4)    1280: (2,9)
    48: (1,5)       384: (1,8)      1296: (2,4,5)
    52: (1,2,3)     400: (1,3,5)    1312: (2,3,6)
    64: (7)         416: (1,2,6)    1536: (1,10)
    72: (3,4)       512: (10)       1568: (1,4,6)
    80: (2,5)       544: (4,6)      1600: (1,3,7)
    96: (1,6)       576: (3,7)      1664: (1,2,8)
   104: (1,2,4)     640: (2,8)      1680: (1,2,3,5)

Strictly increasing runs are counted by A124768.
The normal case is A164894.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly decreasing version is A333256.
Cf. A000120, A029931, A048793, A066099, A070939, A124762, A228351, A333217, A333220, A333379.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Less@@stc[#]&]

A326702 Number of distinct vertices in the set-system with BII-number n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 22 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			The BII-number of {{1,2},{1,4}} is 260, with distinct vertices {1,2,4}, so a(260) = 3.

Positions of first appearances are A072639.

Cf. A000120, A029931, A048793, A070939, A326031, A326701, A326703, A326704.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Union@@bpe/@bpe[n]],{n,0,100}]

A333227 Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97, 99, 101, 102, 103, 105

Offset: 1

Gus Wiseman, Mar 27 2020

nonn

This is the definition used for CoprimeQ in Mathematica.

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 together with the corresponding compositions begins:
   1: (1)          27: (1,2,1,1)      55: (1,2,1,1,1)
   3: (1,1)        28: (1,1,3)        56: (1,1,4)
   5: (2,1)        29: (1,1,2,1)      57: (1,1,3,1)
   6: (1,2)        30: (1,1,1,2)      59: (1,1,2,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    60: (1,1,1,3)
   9: (3,1)        33: (5,1)          61: (1,1,1,2,1)
  11: (2,1,1)      35: (4,1,1)        62: (1,1,1,1,2)
  12: (1,3)        37: (3,2,1)        63: (1,1,1,1,1,1)
  13: (1,2,1)      38: (3,1,2)        65: (6,1)
  14: (1,1,2)      39: (3,1,1,1)      66: (5,2)
  15: (1,1,1,1)    41: (2,3,1)        67: (5,1,1)
  17: (4,1)        44: (2,1,3)        68: (4,3)
  18: (3,2)        47: (2,1,1,1,1)    71: (4,1,1,1)
  19: (3,1,1)      48: (1,5)          72: (3,4)
  20: (2,3)        49: (1,4,1)        75: (3,2,1,1)
  23: (2,1,1,1)    50: (1,3,2)        77: (3,1,2,1)
  24: (1,4)        51: (1,3,1,1)      78: (3,1,1,2)
  25: (1,3,1)      52: (1,2,3)        79: (3,1,1,1,1)

A different ranking of the same compositions is A326675.
Ignoring repeated parts gives A333228.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- q(k) has A124767(k) runs and A333381(k) anti-runs.
- The GCD of q(k) is A326674(k).
- The Heinz number of q(k) is A333219(k).
- The LCM of q(k) is A333226(k).
Coprime or singleton sets are ranked by A087087.
Strict compositions are ranked by A233564.
Constant compositions are ranked by A272919.
Relatively prime compositions appear to be ranked by A291166.
Normal compositions are ranked by A333217.
Cf. A000120, A029931, A048793, A096111, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,120],CoprimeQ@@stc[#]&]

A022290 Replace 2^k in binary expansion of n with Fibonacci(k+2).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 8, 9, 10, 11, 8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19, 13, 14, 15, 16, 16, 17, 18, 19, 18, 19, 20, 21, 21, 22, 23, 24, 21, 22, 23, 24, 24, 25, 26, 27, 26, 27, 28, 29, 29, 30, 31, 32, 21, 22, 23, 24, 24, 25, 26

Offset: 0

Marc LeBrun

			n=4 = 2^2 is replaced by A000045(2+2) = 3. n=5 = 2^2 + 2^0 is replaced by A000045(2+2) + A000045(0+2) = 3+1 = 4. - _R. J. Mathar_, Jan 31 2015
From _Philippe Deléham_, Jun 05 2015: (Start)
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
  0
  1
  2, 3
  3, 4, 5, 6
  5, 6, 7, 8, 8, 9, 10, 11
  8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A054204 (even-indexed Fibonacci numbers), A062877 (odd-indexed Fibonacci numbers), A059590 (factorials), A089625 (primes).

Cf. A003754, A022342, A026274.

a022290 0 = 0
a022290 n = h n 0 $ drop 2 a000045_list where
   h 0 y _      = y
   h x y (f:fs) = h x' (y + f * r) fs where (x',r) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012

A022290 := proc(n)
    dgs := convert(n,base,2) ;
    add( op(i,dgs)*A000045(i+1),i=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015
# second Maple program:
b:= (n, i, j)-> `if`(n=0, 0, j*irem(n, 2, 'q')+b(q, j, i+j)):
a:= n-> b(n, 1$2):
seq(a(n), n=0..127);  # Alois P. Heinz, Jan 26 2022

Table[Reverse[#].Fibonacci[1 + Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 54}] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)

my(m=Mod('x,'x^2-'x-1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,2); \\ Kevin Ryde, Sep 22 2020

def A022290(n):
    a, b, s = 1,2,0
    for i in bin(n)[-1:1:-1]:
        s += int(i)*a
        a, b = b, a+b
    return s # Chai Wah Wu, Sep 10 2022

G.f.: (1/(1-x)) * Sum_{k>=0} F(k+2)*x^2^k/(1+x^2^k), where F = A000045.
a(n) = Sum_{k>=0} A030308(n,k)*A000045(k+2). - Philippe Deléham, Oct 15 2011
a(A003714(n)) = n. - R. J. Mathar, Jan 31 2015
a(A000225(n)) = A001911(n). - Philippe Deléham, Jun 05 2015
From Jeffrey Shallit, Jul 17 2018: (Start)
Can be computed from the recurrence:
a(4*k)   = a(k) + a(2*k),
a(4*k+1) = a(k) + a(2*k+1),
a(4*k+2) = a(k) - a(2*k) + 2*a(2*k+1),
a(4*k+3) = a(k) - 2*a(2*k) + 3*a(2*k+1),
and the initial terms a(0) = 0, a(1) = 1. (End)
a(A003754(n)) = n-1. - Rémy Sigrist, Jan 28 2020
From Rémy Sigrist, Aug 04 2022: (Start)
Empirically:
- a(2*A003714(n)) = A022342(n+1),
- a(3*A003714(n)) = a(4*A003714(n)) = A026274(n) for n > 0.
(End)

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 18 2020

nonn

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 composition (4,3,1,2) has positive subsequence-sums 1, 2, 3, 4, 6, 7, 8, 10, so a(550) = 8.

Dominated by A124770.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702.
Strict knapsack partitions are counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Allowing empty subsequences gives A333257.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A124767, A143823, A295235, A325680, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333257(n) - 1.

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