A242628 -id:A242628 - cp's OEIS frontend

A005940 The Doudna sequence: write n-1 in binary; power of prime(k) in a(n) is # of 1's that are followed by k-1 0's.

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 11, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 13, 22, 33, 28, 55, 42, 63, 40, 77, 70, 105, 60, 175, 90, 135, 48, 121, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 26, 39

Offset: 1

N. J. A. Sloane and J. H. Conway

A permutation of the natural numbers. - Robert G. Wilson v, Feb 22 2005
Fixed points: A029747. - Reinhard Zumkeller, Aug 23 2006
The even bisection, when halved, gives the sequence back. - Antti Karttunen, Jun 28 2014
From Antti Karttunen, Dec 21 2014: (Start)
This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A003961 to the parent, and each child to the right is obtained by doubling the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        5......../ \........6                   9......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       10         15       12         25       18         27       16
  11 14   21  20     35  30   45  24     49  50   75  36    125  54   81  32
etc.
Sequence A163511 is obtained by scanning the same tree level by level, from right to left. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees.
A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 2 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is larger than the right child, A252750 the difference between left and right child for each node from node 2 onward.
(End)
-A008836(a(1+n)) gives the corresponding numerator for A323505(n). - Antti Karttunen, Jan 19 2019
(a(2n+1)-1)/2 [= A244154(n)-1, for n >= 0] is a permutation of the natural numbers. - George Beck and Antti Karttunen, Dec 08 2019
From Peter Munn, Oct 04 2020: (Start)
Each term has the same even part (equivalently, the same 2-adic valuation) as its index.
Using the tree depicted in Antti Karttunen's 2014 comment:
Numbers are on the right branch (4 and descendants) if and only if divisible by the square of their largest prime factor (cf. A070003).
Numbers on the left branch, together with 2, are listed in A102750.
(End)
According to Kutz (1981), he learned of this sequence from American mathematician Byron Leon McAllister (1929-2017) who attributed the invention of the sequence to a graduate student by the name of Doudna (first name Paul?) in the mid-1950's at the University of Wisconsin. - Amiram Eldar, Jun 17 2021
From David James Sycamore, Sep 23 2022: (Start)
Alternative (recursive) definition: If n is a power of 2 then a(n)=n. Otherwise, if 2^j is the greatest power of 2 not exceeding n, and if k = n - 2^j, then a(n) is the least m*a(k) that has not occurred previously, where m is an odd prime.
Example: Use recursion with n = 77 = 2^6 + 13. a(13) = 25 and since 11 is the smallest odd prime m such that m*a(13) has not already occurred (see a(27), a(29),a(45)), then a(77) = 11*25 = 275. (End)
The odd bisection, when transformed by replacing all prime(k)^e in a(2*n - 1) with prime(k-1)^e, returns a(n), and thus gives the sequence back. - David James Sycamore, Sep 28 2022

			From _N. J. A. Sloane_, Aug 22 2022: (Start)
Let c_i = number of 1's in binary expansion of n-1 that have i 0's to their right, and let p(j) = j-th prime.  Then a(n) = Product_i p(i+1)^c_i.
If n=9, n-1 is 1000, c_3 = 1, a(9) = p(4)^1 = 7.
If n=10, n-1 = 1001, c_0 = 1, c_2 = 1, a(10) = p(1)*p(3) = 2*5 = 10.
If n=11, n-1 = 1010, c_1 = 1, c_2 = 1, a(11) = p(2)*p(3) = 15. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..8192 (terms 1..1024 from Reinhard Zumkeller)
Michael De Vlieger, 6 row Doudna Tree diagram as mentioned in Comments.
Michael De Vlieger, Annotated fan-style binary tree showing 10 levels, with a color function where 2^m is shown in medium blue in row m, k < 2^m is darker blue, and k > 2^m is brighter green, with records in each row shown in red.
Ronald E. Kutz, Two unusual sequences, Two-Year College Mathematics Journal, Vol. 12, No. 5 (1981), pp. 316-319.
Index entries for sequences that are permutations of the natural numbers

Cf. A103969. Inverse is A005941 (A156552).
Cf. A125106. [From Franklin T. Adams-Watters, Mar 06 2010]
Cf. A252737 (gives row sums), A252738 (row products), A332979 (largest on row).
Related permutations of positive integers: A163511 (via A054429), A243353 (via A006068), A244154, A253563 (via A122111), A253565, A332977, A334866 (via A225546).
A000120, A003602, A003961, A006519, A053645, A070939, A246278, A250246, A252753, A253552 are used in a formula defining this sequence.
Formulas for f(a(n)) are given for f = A000265, A003963, A007949, A055396, A056239.
Numbers that occur at notable sets of positions in the binary tree representation of the sequence: A000040, A000079, A002110, A070003, A070826, A102750.
Cf. also A000142, A001511, A002450, A112798, A252463, A252464, A252745, A252750, A324054, A324106, A323505, A323508.
Cf. A106737, A290077, A323915, A324052, A324054, A324055, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349 for various number-theoretical sequences applied to (i.e., permuted by) this sequence.
k-adic valuation: A007814 (k=2), A337821 (k=3).
Positions of multiples of 3: A091067.
Primorial deflation: A337376 / A337377.
Sum of prime indices of a(n) is A161511, reverse version A359043.
A048793 lists binary indices, ranked by A019565.
A066099 lists standard comps, partial sums A358134 (ranked by A358170).
Cf. A001222, A029837, A059893, A241916, A242628, A253566, A359042.

a005940 n = f (n - 1) 1 1 where
   f 0 y _          = y
   f x y i | m == 0 = f x' y (i + 1)
           | m == 1 = f x' (y * a000040 i) i
           where (x',m) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
(define (A005940 n) (A005940off0 (- n 1))) ;; The off=1 version, utilizing any one of three different offset-0 implementations:
(definec (A005940off0 n) (cond ((< n 2) (+ 1 n)) (else (* (A000040 (- (A070939 n) (- (A000120 n) 1))) (A005940off0 (A053645 n))))))
(definec (A005940off0 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A003961 (A005940off0 (/ n 2)))) (else (* 2 (A005940off0 (/ (- n 1) 2))))))
(define (A005940off0 n) (let loop ((n n) (i 1) (x 1)) (cond ((zero? n) x) ((even? n) (loop (/ n 2) (+ i 1) x)) (else (loop (/ (- n 1) 2) i (* x (A000040 i)))))))
;; Antti Karttunen, Jun 26 2014

f := proc(n,i,x) option remember ; if n = 0 then x; elif type(n,'even') then procname(n/2,i+1,x) ; else procname((n-1)/2,i,x*ithprime(i)) ; end if; end proc:
A005940 := proc(n) f(n-1,1,1) ; end proc: # R. J. Mathar, Mar 06 2010

f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; Table[ f[n], {n, 67}] (* Robert G. Wilson v, Feb 22 2005 *)
Table[Times@@Prime/@(Join@@Position[Reverse[IntegerDigits[n,2]],1]-Range[DigitCount[n,2,1]]+1),{n,0,100}] (* Gus Wiseman, Dec 28 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, n%2 && (t*=p) || p=nextprime(p+1)); t } \\ M. F. Hasler, Mar 07 2010; update Aug 29 2014

a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); t \\ Charles R Greathouse IV, Nov 11 2021

from sympy import prime
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
print([b(n - 1) for n in range(1, 101)]) # Indranil Ghosh, Apr 10 2017

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A005940(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

From Reinhard Zumkeller, Aug 23 2006, R. J. Mathar, Mar 06 2010: (Start)
a(n) = f(n-1, 1, 1)
  where f(n, i, x) = x if n = 0,
                   = f(n/2, i+1, x) if n > 0 is even
                   = f((n-1)/2, i, x*prime(i)) otherwise. (End)
From Antti Karttunen, Jun 26 2014: (Start)
Define a starting-offset 0 version of this sequence as:
  b(0)=1, b(1)=2, [base cases]
and then compute the rest either with recurrence:
  b(n) = A000040(1+(A070939(n)-A000120(n))) * b(A053645(n)).
or
  b(2n) = A003961(b(n)), b(2n+1) = 2 * b(n). [Compare this to the similar recurrence given for A163511.]
Then define a(n) = b(n-1), where a(n) gives this sequence A005940 with the starting offset 1.
Can be also defined as a composition of related permutations:
a(n+1) = A243353(A006068(n)).
a(n+1) = A163511(A054429(n)). [Compare the scatter plots of this sequence and A163511 to each other.]
This permutation also maps between the partitions as enumerated in the lists A125106 and A112798, providing identities between:
A161511(n) = A056239(a(n+1)). [The corresponding sums ...]
A243499(n) = A003963(a(n+1)). [... and the products of parts of those partitions.]
(End)
From Antti Karttunen, Dec 21 2014  - Jan 04 2015: (Start)
A002110(n) =  a(1+A002450(n)). [Primorials occur at (4^n - 1)/3 in the offset-0 version of the sequence.]
a(n) = A250246(A252753(n-1)).
a(n) = A122111(A253563(n-1)).
For n >= 1, A055396(a(n+1)) = A001511(n).
For n >= 2, a(n) = A246278(1+A253552(n)).
(End)
From Peter Munn, Oct 04 2020: (Start)
A000265(a(n)) = a(A000265(n)) = A003961(a(A003602(n))).
A006519(a(n)) = a(A006519(n)) = A006519(n).
a(n) = A003961(a(A003602(n))) * A006519(n).
A007814(a(n)) = A007814(n).
A007949(a(n)) = A337821(n) = A007814(A003602(n)).
a(n) = A225546(A334866(n-1)).
(End)
a(2n) = 2*a(n), or generally a(2^k*n) = 2^k*a(n). - Amiram Eldar, Oct 03 2022
If n-1 = Sum_{i} 2^(q_i-1), then a(n) = Product_{i} prime(q_i-i+1). These are the Heinz numbers of the rows of A125106. If the offset is changed to 0, the inverse is A156552. - Gus Wiseman, Dec 28 2022

More terms from Robert G. Wilson v, Feb 22 2005
Sign in a formula switched and Maple program added by R. J. Mathar, Mar 06 2010
Binary tree illustration and keyword tabf added by Antti Karttunen, Dec 21 2014

A272020 Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.

Original entry on oeis.org

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

Offset: 0

Peter Kagey, Apr 17 2016

Length of n-th row given by A000120(n);
Min of n-th row given by A001511(n);
Sum of n-th row given by A029931(n);
Product of n-th row given by A096111(n);
Max of n-th row given by A113473(n);
Numerator of sum of reciprocals of n-th row given by A116416(n);
Denominator of sum of reciprocals of n-th row given by A116417(n);
LCM of n-th row given by A271410(n).
The first appearance of n is at A001787(n - 1).
n-th row begins at index A000788(n - 1) for n > 0.
Also the reversed positions of 1's in the reversed binary expansion of n. Also the reversed partial sums of the n-th composition in standard order (row n of A066099). Reversing rows gives A048793. - Gus Wiseman, Jan 17 2023

			Row n is given by the exponents in the binary expansion of 2*n. For example, row 5 = [3, 1] because 2*5 = 2^3 + 2^1.
Row 0: []
Row 1: [1]
Row 2: [2]
Row 3: [2, 1]
Row 4: [3]
Row 5: [3, 1]
Row 6: [3, 2]
Row 7: [3, 2, 1]

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000120, A096111, A116416, A116417, A271410, A001787.
Cf. A048793 gives the rows in reverse order.
Cf. A272011.
Lasts are A001511.
Heinz numbers of the rows are A019565.
Firsts are A029837 or A070939 or A113473.
Row sums are A029931.
A066099 lists standard comps, partial sums A358134, weighted sum A359042.
Cf. A005940, A059893, A125106, A161511, A242628.

T:= proc(n) local i, l, m; l:= NULL; m:= n;
      if n=0 then return [][] fi; for i while m>0 do
      if irem(m, 2, 'm')=1 then l:=i, l fi od; l
    end:
seq(T(n), n=0..35);  # Alois P. Heinz, Nov 27 2024

Table[Reverse[Join@@Position[Reverse[IntegerDigits[n,2]],1]],{n,0,100}] (* Gus Wiseman, Jan 17 2023 *)

A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is lexicographic (see example). - Joerg Arndt, Sep 03 2013
The equivalent sequence for compositions (ordered partitions) is A228369. - Omar E. Pol, Oct 19 2019

			First six rows are:
[[1]];
[[1, 1], [2]];
[[1, 1, 1], [1, 2], [3]];
[[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
----------------------------------
.                     Ordered
n  j      Diagram     partition j
----------------------------------
.               _
1  1           |_|    1;
.             _ _
2  1         | |_|    1, 1,
2  2         |_ _|    2;
.           _ _ _
3  1       | | |_|    1, 1, 1,
3  2       | |_ _|    1, 2,
3  3       |_ _ _|    3;
.         _ _ _ _
4  1     | | | |_|    1, 1, 1, 1,
4  2     | | |_ _|    1, 1, 2,
4  3     | |_ _ _|    1, 3,
4  4     |   |_ _|    2, 2,
4  5     |_ _ _ _|    4;
...
(End)

Alois P. Heinz, Rows n = 1..19, flattened
Wikiversity, Lexicographic and colexicographic order

Row lengths are given in A006128.
Partition lengths are in A193173.
Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A196931.
Partition maxima are A194546.
The reflected version is A211992.
The length-sensitive version (sum/length/lex) is A036036.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A193073.
Compositions under the same ordering (sum/lex) are A228369.
The reverse-lexicographic version (sum/revlex) is A228531.
The Heinz numbers of these partitions are A334437.
Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, l[]
          else seq(b(n-i, [l[], i]), i=`if`(l=[],1,l[-1])..n)
            fi
          end;
      ll:= NULL; b(n, []); ll
    end:
seq(T(n), n=1..8);  # Alois P. Heinz, Jul 16 2011

T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Aug 05 2015, after Alois P. Heinz *)
Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* Robert Price, May 18 2020 *)

t = [[[]]]
for n in range(1, 10):
    p = []
    for minp in range(1, n):
        p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
    t.append(p + [[n]])
print(t)
# Andrey Zabolotskiy, Oct 18 2019

A253565 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 8, 7, 25, 15, 27, 10, 18, 12, 16, 11, 49, 35, 125, 21, 75, 45, 81, 14, 50, 30, 54, 20, 36, 24, 32, 13, 121, 77, 343, 55, 245, 175, 625, 33, 147, 105, 375, 63, 225, 135, 243, 22, 98, 70, 250, 42, 150, 90, 162, 28, 100, 60, 108, 40, 72, 48, 64, 17, 169, 143, 1331, 91, 847, 539, 2401, 65, 605, 385, 1715, 275, 1225, 875, 3125, 39

Offset: 0

Antti Karttunen, Jan 03 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A253550 to the parent, and each child to the right is obtained by applying A253560 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........9                   6......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       25         15       27         10       18         12       16
11 49   35  125    21  75   45  81     14  50   30  54     20  36   24  32
etc.
Sequence A253563 is the mirror image of the same tree. Also in binary trees A005940 and A163511 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees. Of these four trees, this is the one where the left child is always smaller than the right child.
Note that the indexing of sequence starts from 0, although its range starts from one.
The term a(n) is the Heinz number of the adjusted partial sums of the n-th composition in standard order, where (1) 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, (2) the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and (3) we define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts. See formula for a simplification. A triangular form is A242628. The inverse is A253566. The non-adjusted version is A358170. - Gus Wiseman, Dec 17 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between compositions and partitions. The n-th composition in standard order together with the reversed prime indices of a(n) are:
   0:        () -> ()
   1:       (1) -> (1)
   2:       (2) -> (2)
   3:     (1,1) -> (1,1)
   4:       (3) -> (3)
   5:     (2,1) -> (2,2)
   6:     (1,2) -> (2,1)
   7:   (1,1,1) -> (1,1,1)
   8:       (4) -> (4)
   9:     (3,1) -> (3,3)
  10:     (2,2) -> (3,2)
  11:   (2,1,1) -> (2,2,2)
  12:     (1,3) -> (3,1)
  13:   (1,2,1) -> (2,2,1)
  14:   (1,1,2) -> (2,1,1)
  15: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A253566.
Cf. A252737 (row sums), A252738 (row products).
Cf. A122111, A163511, A253550, A253560, A252464, A253563, A054429.
Applying A001222 gives A000120.
A reverse version is A005940.
These are the Heinz numbers of the rows of A242628.
Sum of prime indices of a(n) is A359043, reverse A161511.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A059893, A070939, A241916, A358135, A358137, A358195, A359042.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]-1]+1,{n,0,60}] (* Gus Wiseman, Dec 17 2022 *)

a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).
As a composition of related permutations:
a(n) = A122111(A163511(n)).
a(n) = A253563(A054429(n)).
Other identities and observations. For all n >= 0:
a(2n+1) - a(2n) > 0. [See the comment above.]
If n = 2^(x_1)+...+2^(x_k) then a(n) = Product_{i=1..k} prime(x_k-x_{i-1}-k+i) where x_0 = 0. - Gus Wiseman, Dec 23 2022

A161511 Number of 1...0 pairs in the binary representation of 2n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 5, 5, 6, 5, 7, 6, 7, 5, 8, 7, 8, 6, 9, 7, 8, 5, 6, 6, 7, 6, 8, 7, 8, 6, 9, 8, 9, 7, 10, 8, 9, 6, 10, 9, 10, 8, 11, 9, 10, 7, 12, 10, 11, 8, 12, 9, 10, 6, 7, 7, 8, 7, 9, 8, 9, 7, 10, 9, 10, 8, 11, 9, 10, 7, 11, 10, 11, 9, 12, 10, 11, 8, 13, 11, 12, 9, 13

Offset: 0

Franklin T. Adams-Watters, Jun 11 2009

Row (partition) sums of A125106.

a(n) is also the weight (= sum of parts) of the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

			For n = 5, the binary representation of 2n is 1010; the 1...0 pairs are 10xx, 1xx0, and xx10, so a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Peter J. Taylor, Recursion for A161511, answer to question on MathOverflow (2025).

Cf. A000120, A243499 (gives the corresponding products), A227183, A056239, A243503, A006068, A163511.
Sum of prime indices of A005940.
Row sums of A125106.
A reverse version is A359043, row sums of A242628.
A029837 adds up standard compositions, row sums of A066099.
A029931 adds up binary indices, row sums of A048793.
Cf. A059893, A228351, A230877, A231204, A253565, A358194, A359042, A359359.

a[0] = 0; a[n_] := If[EvenQ[n], a[n/2] + DigitCount[n/2, 2, 1], a[(n-1)/2] + 1]; Array[a, 93, 0] (* Jean-François Alcover, Sep 09 2017 *)

a(n)=local(t,k);t=0;k=1;while(n>0,if(n%2==0,k++,t+=k);n\=2);t

def A161511(n):
    a, b = 0, 0
    for i, j in enumerate(bin(n)[:1:-1], 1):
        if int(j):
            a += i-b
            b += 1
    return a # Chai Wah Wu, Jul 26 2023

;; Two variants, the recursive one requiring memoizing definec-macro from Antti Karttunen's IntSeq-library.
(define (A161511 n) (let loop ((n n) (i 1) (s 0)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ i 1) s)) (else (loop (/ (- n 1) 2) i (+ s i))))))
(definec (A161511 n) (cond ((zero? n) n) ((even? n) (+ (A000120 n) (A161511 (/ n 2)))) (else (+ 1 (A161511 (/ (- n 1) 2))))))
;; Antti Karttunen, Jun 28 2014

a(0) = 0; a(2n) = a(n) + A000120(n); a(2n+1) = a(n) + 1.
From Antti Karttunen, Jun 28 2014: (Start)
Can be also obtained by mapping with an appropriate permutation from the lists of partition sizes computed for other enumerations similar to A125106:
a(n) = A227183(A006068(n)).
a(n) = A056239(A005940(n+1)).
a(n) = A243503(A163511(n)). (End)
a(n) = A029931(n) - binomial(A000120(n),2). - Gus Wiseman, Jan 03 2023
a(n) = a(n - A048896(n-1)) + 1 for n>=1 (see Peter J. Taylor link). - Mikhail Kurkov, Jul 04 2025

A125106 Enumeration of partitions by binary representation: each 1 is a part; the part size is 1 more than the number of 0's in the rest of the number.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Dec 10 2006

Another way to describe this: starting with the binary representation and a counter set at one, count the 0's from right to left. Write a term equal to the counter for each "1" encountered.
A101211 is a similar sequence, with A005811 elements per row which maps natural numbers to compositions (ordered partitions).
There are two ways to consider this as a table: taking each partition as a row, or taking the partitions generated by 2^(n-1) through 2^n-1 as a row.
Taking the n-th row as multiple partitions, it consists of those partitions with the first hook size (largest part plus number of parts minus 1) equal to n. The number of integers in this n-th row is A001792(n-1), and the row sum is A049611.
Taking each partition as a separate row, the row lengths are A000120, and the row sums are A161511.
Heinz numbers of the rows are A005940. - Gus Wiseman, Jan 17 2023

			Row 4:
1000 [4]
1001 [3,1]
1010 [3,2]
1011 [2,1,1]
1100 [3,3]
1101 [2,2,1]
1110 [2,2,2]
1111 [1,1,1,1]

Alois P. Heinz, Rows n = 1..12, flattened

Cf. A000041, A005811, A037016, A101211, A001792, A049611, A126411.
Each partition as row: A000120 (row widths), A161511 (row sums), A243499 (row products).
Cf. A005940. - Franklin T. Adams-Watters, Mar 06 2010
Lasts are A001511.
Firsts are A008687.
Cf. A019565, A029837, A029931, A048793, A059893, A066099, A070939, A242628.

b:= proc(n) local c, l, m; l:=[][]; m:= n; c:=1;
      while m>0 do if irem(m, 2, 'm')=0 then c:= c+1
         else l:= c, l fi
      od; l
    end:
T:= n-> seq(b(i), i=2^(n-1)..2^n-1):
seq(T(n), n=1..7);  # Alois P. Heinz, Sep 25 2015

f[k_] := (bits = IntegerDigits[k, 2]; zerosCount = Reverse[ Accumulate[ 1-Reverse[bits] ] ] + 1; Select[ Transpose[ {bits, zerosCount} ], First[#] == 1 & ][[All, 2]]); row[n_] := Table[ f[k], {k, 2^(n-1), 2^n-1}]; Flatten[ Table[ row[n], {n, 1, 5}]] (* Jean-François Alcover, Jan 24 2012 *)
scc[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Reverse[scc[n]-Range[Length[scc[n]]]+1],{n,0,20}] (* Gus Wiseman, Jan 17 2023 *)

Partition 2n is partition n with every part size increased by 1; partition 2n+1 is partition n with an additional part of size 1.

T(n,k) = A272020(n,k) - A000120(n) + k. - Gus Wiseman, Jan 17 2023

Edited by Franklin T. Adams-Watters, Jun 11 2009

cp's OEIS Frontend

A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

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A241597 Number of compositions corresponding to the n-th partition in A241596 (or A242628).

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A005940 The Doudna sequence: write n-1 in binary; power of prime(k) in a(n) is # of 1's that are followed by k-1 0's.

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A272020 Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.

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A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

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A253565 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).

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A161511 Number of 1...0 pairs in the binary representation of 2n.

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