A073137 -id:A073137 - cp's OEIS frontend

A073138 Largest number having in its binary representation the same number of 0's and 1's as n.

0, 1, 2, 3, 4, 6, 6, 7, 8, 12, 12, 14, 12, 14, 14, 15, 16, 24, 24, 28, 24, 28, 28, 30, 24, 28, 28, 30, 28, 30, 30, 31, 32, 48, 48, 56, 48, 56, 56, 60, 48, 56, 56, 60, 56, 60, 60, 62, 48, 56, 56, 60, 56, 60, 60, 62, 56, 60, 60, 62, 60, 62, 62, 63, 64, 96, 96, 112, 96, 112, 112

Offset: 0

Reinhard Zumkeller, Jul 16 2002

From Trevor Green (green(AT)snoopy.usask.ca), Nov 26 2003: (Start)
a(n)/n has an accumulation point at x exactly when x is in the interval [1, 2]. Proof: Clearly n <= a(n) < 2n. Let b(n) = a(n)/n, then b(n) must always lie in [1,2) and all the accumulation points of the sequence must lie in [1,2]. We shall show that every such number is an accumulation point.
First, consider any d-bit integer n. Suppose that z of these bits are 0. Let n' be the (d+z)-bit integer whose first d bits are the same as those of n and whose remaining bits are all 1. Then a(n') will have to be the (d+z)-bit integer whose first d bits are all 1 and whose last z bits are all 0.
Thus n' = (n+1)*2^z-1; a(n') = (2^d-1)2^z; and b(n') = (2^d-1)/(n+1) + epsilon, where 0 < epsilon < 2^(1-d). So to get an accumulation point x, we just choose n(d) to be the d-bit integer such that (2^d-1)/(n(d)+1) < x <= (2^d-1)/n(d), or equivalently, n(d) = floor((2^d-1)/x). If x lies in [1,2), then n(d) will always be a d-bit number for sufficiently large d.
Then n'(d) yields an increasing subsequence of the integers for which b(n'(d)) converges to x. For x = 2, choose n(d) = 2^(d-1), which is always a d-bit number; then b(n'(d)) = (2^d-1)/(2^(d-1)+1) + epsilon = 2 + epsilon', where epsilon' also heads for 0 as d blows up. This proves the claim.
(End)

			a(20)=24, as 20='10100' and 24 is the greatest number having two 1's and three 0's: 17='10001', 18='10010', 20='10100' and 24='11000'.

Seiichi Manyama, Table of n, a(n) for n = 0..8191 (terms 0..1023 from T. D. Noe)
Index entries for sequences related to binary expansion of n

Cf. A007088, A073137, A000523, A073139, A073140, A073141, A319650.
Cf. A030109.
Cf. A038573.
Decimal equivalent of A221714. - N. J. A. Sloane, Jan 26 2013

a073138 n = a038573 n * a080100 n  -- Reinhard Zumkeller, Jan 16 2012

a:= n-> Bits[Join](sort(Bits[Split](n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 26 2021

f[n_] := Module[{idn=IntegerDigits[n, 2], o, l}, l=Length[idn]; o=Count[idn, 1]; FromDigits[Join[Table[1, {o}], Table[0, {l-o}]], 2]]; Table[f[i], {i, 0, 70}]
ln[n_] := Module[{idn=IntegerDigits[n, 2], len, zer}, len=Length[idn]; zer=Count[idn, 0]; FromDigits[Join[Table[1, {len-zer}], Table[0, {zer}]], 2]]; Table[ln[i], {i, 0, 70}]
a[z_] := 2^(Floor[Log[2, z]] + 1) * (1 - 2^(-Sum[k, {k, IntegerDigits[n, 2]}])) Column[Table[a[p], {p, 500}], Right] (* Trevor G. Hyde (thyde12(AT)amherst.edu), Jul 14 2008 *)
Table[FromDigits[ReverseSort[IntegerDigits[n,2]],2],{n,0,70}] (* Harvey P. Dale, Mar 13 2023 *)

a(n) = fromdigits(vecsort(binary(n),,4), 2); \\ Michel Marcus, Sep 26 2018

def a(n): return int("".join(sorted(bin(n)[2:], reverse=True)), 2)
print([a(n) for n in range(71)]) # Michael S. Branicky, Jun 27 2021

def A073138(n): return (m:=1<>n.bit_count()) # Chai Wah Wu, Aug 18 2025

a(n+1) = a(floor(n/2))*2 + (n mod 2)*(2^floor(log_2(n)) - a(floor(n/2))); a(0)=0.
A023416(a(n)) = A023416(n), A000120(a(n)) = A000120(n).
a(0)=0, a(1)=1, a(2n) = 2a(n), a(2n+1) = a(n) + 2^floor(log_2(n)). - Ralf Stephan, Oct 05 2003
a(n) = 2^(floor(log_2(n)) + 1) * (1 - 2^(-d(n))) where d(n) = digit sum of base-2 expansion of n. - Trevor G. Hyde (thyde12(AT)amherst.edu), Jul 14 2008
a(n) = A038573(n) * A080100(n). - Reinhard Zumkeller, Jan 16 2012
n <= a(n) < 2n. - Charles R Greathouse IV, Aug 07 2024

A099627 Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 23, 31, 32, 33, 35, 39, 47, 63, 64, 65, 67, 71, 79, 95, 127, 128, 129, 131, 135, 143, 159, 191, 255, 256, 257, 259, 263, 271, 287, 319, 383, 511, 512, 513, 515, 519, 527, 543, 575, 639, 767, 1023, 1024, 1025, 1027, 1031, 1039

Offset: 0

Henry Bottomley, Oct 25 2004

Positive integers m where m-th Catalan number A000108(m) = C(2m,m)/(m+1) is not divisible by 4, i.e. where A048881(m) is 0 or 1.
Numbers in A000225 or A099628.
From Charles L. Hohn, Jul 25 2024: (Start)
Integers >=1 whose binary digit counts (number of 0s and number of 1s) are distinct from those of any smaller number.
Binary analog of A179239 for n>=1.
All integers whose binary expression conforms to regex /^10*1*$/, shown in base 10 in ascending numeric order. (End)
Together with 0 all fixed points of A073137. - Alois P. Heinz, Jan 30 2025

			Triangle starts:                  In binary:
   k = 0  1  2  3  4  5
n
0      1                               1
1      2  3                           10     11
2      4  5  7                       100    101    111
3      8  9 11 15                   1000   1001   1011   1111
4     16 17 19 23 31               10000  10001  10011  10111  11111
5     32 33 35 39 47 63           100000 100001 100011 100111 101111 111111
E.g. T(5,3) = 2^5 + 2^3-1 = 32 + 7 = 39 (100111 in binary).

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

A053221 (row sums), A000079 (left diagonal), A000225 (right diagonal).
A048645 (see formula).
Partial sums of A232089.
Cf. A073137, A179239.

a099627 n k = a099627_tabl !! n !! k
a099627_row n = a099627_tabl !! n
a099627_tabl = iterate (\xs@(x:_) -> (2 * x) : map ((+ 1) . (* 2)) xs) [1]
-- Reinhard Zumkeller, Dec 19 2012

Table[2^n+2^k -1,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 27 2016 *)

As sequence, a(n) = A048645(n+2) - 1.

G.f.: (1 - x - x^2*y)/((1 - x)*(1 - 2*x)*(1 - x*y)*(1 - 2*x*y)). - Stefano Spezia, Aug 11 2024

A073139 Difference between the largest and smallest number having in binary representation the same number of 0's and 1's as n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 7, 7, 9, 7, 9, 9, 7, 7, 9, 9, 7, 9, 7, 7, 0, 0, 15, 15, 21, 15, 21, 21, 21, 15, 21, 21, 21, 21, 21, 21, 15, 15, 21, 21, 21, 21, 21, 21, 15, 21, 21, 21, 15, 21, 15, 15, 0, 0, 31, 31, 45, 31, 45, 45, 49, 31, 45, 45, 49, 45, 49, 49, 45, 31

Offset: 0

Reinhard Zumkeller, Jul 16 2002

nonn

a(n) = A073138(n) - A073137(n).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A007088, A073140, A073141.

d[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[Sort[idn2,#1>#2&],2]- FromDigits[ RotateRight[Sort[idn2],1],2]]; Array[d,90,0] (* Harvey P. Dale, Oct 22 2011 *)

A073141 Product of the largest and smallest number having in binary representation the same number of 0's and 1's as n.

Original entry on oeis.org

0, 1, 4, 9, 16, 30, 30, 49, 64, 108, 108, 154, 108, 154, 154, 225, 256, 408, 408, 532, 408, 532, 532, 690, 408, 532, 532, 690, 532, 690, 690, 961, 1024, 1584, 1584, 1960, 1584, 1960, 1960, 2340, 1584, 1960, 1960, 2340, 1960, 2340, 2340, 2914, 1584, 1960

Offset: 0

Reinhard Zumkeller, Jul 16 2002

nonn

a(n) = A073137(n) * A073138(n).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A007088, A073139, A073140.

pls[n_]:=With[{d=FromDigits[#,2]&/@Select[Permutations[IntegerDigits[n,2]],#[[1]]==1&]},Max[d]Min[d]]; Join[{0},Array[pls,50]] (* Harvey P. Dale, May 29 2025 *)

A073140 Sum of the largest and smallest number having in binary representation the same number of 0's and 1's as n.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 11, 14, 16, 21, 21, 25, 21, 25, 25, 30, 32, 41, 41, 47, 41, 47, 47, 53, 41, 47, 47, 53, 47, 53, 53, 62, 64, 81, 81, 91, 81, 91, 91, 99, 81, 91, 91, 99, 91, 99, 99, 109, 81, 91, 91, 99, 91, 99, 99, 109, 91, 99, 99, 109, 99, 109, 109, 126, 128, 161, 161, 179

Offset: 0

Reinhard Zumkeller, Jul 16 2002

nonn

a(n) = A073137(n) + A073138(n).

Index entries for sequences related to binary expansion of n

Cf. A007088, A073139, A073141.

A361479 a(n) is the least integer whose binary expansion has the same multiset of run-lengths as that of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 7, 8, 9, 10, 9, 12, 9, 8, 15, 16, 17, 18, 19, 18, 21, 18, 17, 24, 19, 18, 19, 24, 17, 16, 31, 32, 33, 34, 35, 36, 37, 36, 35, 34, 37, 42, 37, 36, 37, 34, 33, 48, 35, 36, 51, 36, 37, 36, 35, 56, 35, 34, 35, 48, 33, 32, 63, 64, 65, 66, 67

Offset: 0

Rémy Sigrist, Mar 13 2023

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A073137, A361478, A361480.

a(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r=vecsort(r); my (h=0, t=#r+1, v=0); for (k=1, #r, v=(v+k%2)*2^if (k%2, r[h++], r[t--])-k%2;); return (v); }

a(a(n)) = a(n).

a(n) <= n with equality iff n = 0 or belongs to A175020.

A375216 n is the a(n)-th nonnegative integer having its multiset of binary digits.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jan 30 2025

			a(26) = 5 because 26 is the 5th nonnegative integer having its multiset of binary digits: 19 = 10011_2, 21 = 10101_2, 22 = 10110_2, 25 = 11001_2, 26 = 11010_2.

Alois P. Heinz, Table of n, a(n) for n = 0..65536

Cf. A073137.

p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= sort(Bits[Split](n)); p(t):= p(t)+1
    end:
seq(a(n), n=0..100);

Ordinal transform of A073137.

cp's OEIS Frontend

A073138 Largest number having in its binary representation the same number of 0's and 1's as n.

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A099627 Triangle read by rows: T(n,k) = 2^n + 2^k - 1 with n >= k >= 0.

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A073139 Difference between the largest and smallest number having in binary representation the same number of 0's and 1's as n.

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A073141 Product of the largest and smallest number having in binary representation the same number of 0's and 1's as n.

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A073140 Sum of the largest and smallest number having in binary representation the same number of 0's and 1's as n.

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A361479 a(n) is the least integer whose binary expansion has the same multiset of run-lengths as that of n.

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PARI

Formula

A375216 n is the a(n)-th nonnegative integer having its multiset of binary digits.

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