A073138 -id:A073138 - cp's OEIS frontend

A319785 a(n) = A073138(n) + A038573(n).

0, 2, 3, 6, 5, 9, 9, 14, 9, 15, 15, 21, 15, 21, 21, 30, 17, 27, 27, 35, 27, 35, 35, 45, 27, 35, 35, 45, 35, 45, 45, 62, 33, 51, 51, 63, 51, 63, 63, 75, 51, 63, 63, 75, 63, 75, 75, 93, 51, 63, 63, 75, 63, 75, 75, 93, 63, 75, 75, 93, 75, 93, 93, 126, 65, 99, 99, 119

Offset: 0

Seiichi Manyama, Sep 27 2018

This sequence is different from A055944. For example, A055944(5) = 10 and a(5) = 9.

Seiichi Manyama, Table of n, a(n) for n = 0..8191

Base b: this sequence (b=2), A319803 (b=3), A319804 (b=4), A319805 (b=5), A319806 (b=6), A319807 (b=7), A319808 (b=8), A319747 (b=9), A052008 (b=10).

Cf. A038573, A055944, A073138.

Table[FromDigits[Reverse[#], 2] + FromDigits[#, 2] & [Sort[IntegerDigits[n, 2]]], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = my(dn=binary(n)); fromdigits(vecsort(dn), 2) + fromdigits(vecsort(dn,,4), 2); \\ Michel Marcus, Sep 28 2018

A319650 a(n) = A073138(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 3, 0, 1, 0, 0, 0, 7, 6, 9, 4, 7, 6, 7, 0, 3, 2, 3, 0, 1, 0, 0, 0, 15, 14, 21, 12, 19, 18, 21, 8, 15, 14, 17, 12, 15, 14, 15, 0, 7, 6, 9, 4, 7, 6, 7, 0, 3, 2, 3, 0, 1, 0, 0, 0, 31, 30, 45, 28, 43, 42, 49, 24, 39, 38, 45, 36, 43, 42, 45, 16

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..8191

Cf. A073138, A164884.

a(n) = fromdigits(vecsort(binary(n),,4), 2) - n; \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k) - i}
end
p A(2, 100)

A380544 Numbers of the form A073138(k) XOR A038573(k).

Original entry on oeis.org

0, 3, 5, 9, 15, 17, 27, 33, 51, 63, 65, 99, 119, 129, 195, 231, 255, 257, 387, 455, 495, 513, 771, 903, 975, 1023, 1025, 1539, 1799, 1935, 2015, 2049, 3075, 3591, 3855, 3999, 4095, 4097, 6147, 7175, 7695, 7967, 8127, 8193, 12291, 14343, 15375, 15903, 16191, 16383, 16385

Offset: 1

Frederik P.J. Vandecasteele, Jun 23 2025

The plot of the sequence has a blancmange appearance. The discontinuities in the curve are at a(n) = k^2 + 1.
The bit length of a(n+1) appears to be A000267(n)+1.
The number of 0-bits in the binary expansion of a(n+1) appears to be A082375(n).

			k = 17. A073138(17) = 24 = 11000_2. A038573(17) = 3 = 00011_2. 11000_2 XOR 00011_2 = 11011_2 = 27. 27 is a term.

Cf. A073138, A038573, A000267, A082375.

Subsequence of A006995.

s[n_, k_] := (2^n-1)*(2^(k-n)+1); Join[{0}, Table[s[n, k], {k, 2, 15}, {n, 1, Floor[k/2]}] // Flatten] (* Amiram Eldar, Jun 23 2025 *)

a8(n) = fromdigits(vecsort(binary(n), , 4), 2);
a3(n) = 2^hammingweight(n)-1;
lista(nn) = Set(vector(nn, n, bitxor(a3(n), a8(n)))); \\ Michel Marcus, Jun 23 2025

More terms from Michel Marcus, Jun 23 2025

A004086 Read n backwards (referred to as R(n) in many sequences).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Also called digit reversal of n.

Leading zeros (after the reversal has taken place) are omitted. - N. J. A. Sloane, Jan 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..50000 (first 1001 terms from Franklin T. Adams-Watters)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], Dec 08 2020.
Michael Penn, A digit moving number puzzle., YouTube video, 2022.

Cf. A004185, A004186, A009996, A073138, A188649, A221714.

Cf. A030101, A030102, A030103, A030104, A030105, A030107, A030108.

a004086 = read . reverse . show  -- Reinhard Zumkeller, Apr 11 2011

|.&.": i.@- 1e5 NB. Stephen Makdisi, May 14 2018

read transforms; A004086 := digrev; #cf "Transforms" link at bottom of page
A004086:=proc(n) local s,t; if n<10 then n else s:=irem(n,10,'t'); while t>9 do s:=s*10+irem(t,10,'t') od: s*10+t fi end; # M. F. Hasler, Jan 29 2012

Table[FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 75}]
IntegerReverse[Range[0,80]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)

dig(n) = {local(m=n,r=[]); while(m>0,r=concat(m%10,r);m=floor(m/10));r}
A004086(n) = {local(b,m,r);r=0;b=1;m=dig(n);for(i=1,matsize(m)[2],r=r+b*m[i];b=b*10);r} \\ Michael B. Porter, Oct 16 2009

A004086(n)=fromdigits(Vecrev(digits(n))) \\ M. F. Hasler, Nov 11 2010, updated May 11 2015, Sep 13 2019

def A004086(n):
    return int(str(n)[::-1]) # Chai Wah Wu, Aug 30 2014

For n > 0, a(a(n)) = n iff n mod 10 != 0. - Reinhard Zumkeller, Mar 10 2002
a(n) = d(n,0) with d(n,r) = r if n=0, otherwise d(floor(n/10), r*10+(n mod 10)). - Reinhard Zumkeller, Mar 04 2010
a(10*n+x) = x*10^m + a(n) if 10^(m-1) <= n < 10^m and 0 <= x <= 9. - Robert Israel, Jun 11 2015

Extended by Ray Chandler, Dec 30 2004

A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 12, 22, 23, 24, 25, 26, 27, 28, 29, 3, 13, 23, 33, 34, 35, 36, 37, 38, 39, 4, 14, 24, 34, 44, 45, 46, 47, 48, 49, 5, 15, 25, 35, 45, 55, 56, 57, 58, 59, 6, 16, 26, 36, 46, 56, 66, 67, 68, 69, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Record values: A009994. - Reinhard Zumkeller, Dec 05 2009
If we define "sortable primes" as prime numbers that remain prime when their digits are sorted in increasing order, then all absolute primes (A003459) are sortable primes but not all sortable primes are absolute primes. For example, 311 is both sortable and absolute, and 271 is sortable but not absolute, since its digits can be permuted to 217 = 7 * 31 or 712 = 2^3 * 89, etc. - Alonso del Arte, Oct 05 2013
The above mentioned "sortable primes" are listed in A211654, the nontrivial ones (with digits not in nondecreasing order) in A086042. - M. F. Hasler, Jul 30 2019

			a(19) = 19 because the digits are already in increasing order.
a(20) = 2 because the digits of 20 are 2 and 0, which in increasing order are 0 and 2, but since zero-padding is not allowed on the left, the zero digit is dropped and we are left with 2.
a(21) = 12 because the digits of 21 are 2 and 1, which in increasing order are 1 and 2.

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

Cf. A004719, A004151, A004086, A004186, A009996, A221714, A073138, A193581, A193582, A065641.

Cf. A211654 (sortable primes) and subsequence A086042 (nontrivial solutions).

import Data.List (sort)
a004185 n = read $ sort $ show n :: Integer
-- Reinhard Zumkeller, Aug 10 2011

A004185:=func; [n eq 0 select 0 else A004185(n): n in [0..57]]; // Bruno Berselli, Apr 03 2012

A004185 := proc(n)
    local dgs;
    convert(n,base,10) ;
    dgs := sort(%,`>`) ;
    add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
seq(A004185(n),n=0..20) ; # R. J. Mathar, Jul 26 2015

FromDigits[Sort[DeleteCases[IntegerDigits[#], 0]]]&/@Range[0, 60] (* Harvey P. Dale, Nov 29 2011 *)

a(n)=fromdigits(vecsort(digits(n))) \\ Charles R Greathouse IV, Feb 06 2017

def A004185(n):
    return int(''.join(sorted(str(n))).replace('0','')) if n > 0 else 0 # Chai Wah Wu, Nov 10 2015

A038573 a(n) = 2^A000120(n) - 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31

Offset: 0

Marc LeBrun

Essentially the same sequence as A001316, which has much more information, and also A159913. - N. J. A. Sloane, Jun 05 2009
Smallest number with same number of 1's in its binary expansion as n.
Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v, Jan 24 2006
From Gary W. Adamson, Jun 04 2009: (Start)
As an infinite string, 2^n terms per row starting with "1": (1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)
Row sums of that triangle = A027649: (1, 4, 14, 46, 454, ...); where the next row sum = current term of A027649 + next term in finite difference row of A027649, i.e., (1, 3, 10, 32, 100, 308, ...) = A053581. (End)
From Omar E. Pol, Jan 24 2016: (Start)
Partial sums give A267700.
a(n) is also the number of cells turned ON at n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.
a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)
Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

			9 = 1001 -> 0011 -> 3, so a(9)=3.
From _Gary W. Adamson_, Jun 04 2009: (Start)
Triangle read by rows:
  1;
  1, 3;
  1, 3, 3, 7;
  1, 3, 3, 7, 3, 7, 7, 15;
  1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;
  ...
Row sums: (1, 4, 14, 46, ...) = A027649 = last row terms + new set of terms such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(2) + A053581(3). (End)
The rows of this triangle converge to A159913. - _N. J. A. Sloane_, Jun 05 2009
G.f. = x + x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 7*x^7 + x^8 + 3*x^9 + 3*x^10 + 7*x^11 + ... - _Michael Somos_, Jul 24 2023

Seiichi Manyama, Table of n, a(n) for n = 0..8191 (terms 0..1023 from T. D. Noe)
David Applegate, The movie version
Michael Gilleland, Some Self-Similar Integer Sequences
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences
Index entries for sequences that are fixed points of mappings

Cf. A007313, A115378, A073138.
This is Guy Steele's sequence GS(3, 6) (see A135416).
Cf. also A000079, A001316, A027649, A053581, A159913, A267700.
Write n in b-ary, sort digits into increasing order: this sequence (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).
Column k=0 of A340666.

a038573 0 = 0
a038573 n = (m + 1) * (a038573 n') + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 10 2012, Feb 07 2011
(Python 3.10+)
def A038573(n): return (1<Chai Wah Wu, Nov 15 2022

seq(2^convert(convert(n,base,2),`+`)-1, n=0..100); # Robert Israel, Jan 24 2016

Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]
Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 24 2006 *)

{a(n) = 2^subst(Pol(binary(n)), x, 1) - 1};

a(n) = 2^hammingweight(n)-1; \\ Michel Marcus, Jan 24 2016

a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n) - 1. - Daniele Parisse
a(n) = number of positive integers k < n such that n XOR k = n-k (cf. A115378). - Paul D. Hanna, Jan 21 2006
a(n) = f(n, 1) with f(x, y) = if x = 0 then y - 1 else f(floor(x/2), y*(1 + x mod 2)). - Reinhard Zumkeller, Nov 21 2009
a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012
a(n) = Sum_{i=1..n} C(n,i) mod 2. - Wesley Ivan Hurt, Nov 17 2017
G.f.: -1/(1 - x) + Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Aug 20 2019
G.f. A(x) = x + x^2*A(x) + (1 + 2*x)*(1 - x^2)*A(x^2). - Michael Somos, Jul 24 2023

More terms from Erich Friedman

New definition from N. J. A. Sloane, Mar 01 2008

A256999 a(n)=n for n <= 1; for n >= 2, a(n) = largest number that can be obtained by rotating non-msb bits of binary expansion of n (with A080541 or A080542).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 6, 7, 8, 12, 12, 14, 12, 14, 14, 15, 16, 24, 24, 28, 24, 26, 28, 30, 24, 28, 26, 30, 28, 30, 30, 31, 32, 48, 48, 56, 48, 52, 56, 60, 48, 52, 52, 58, 56, 58, 60, 62, 48, 56, 52, 60, 52, 58, 58, 62, 56, 60, 58, 62, 60, 62, 62, 63, 64, 96, 96, 112, 96, 104, 112, 120, 96, 100, 104, 114, 112, 116, 120, 124, 96, 104, 100

Offset: 0

Antti Karttunen, May 16 2015

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A080541, A080542, A073138, A246593, A257697.
Cf. A257250 (fixed points of this sequence).
Cf. also A163380 (analogous sequence when rotating all bits of binary representation).

A319722 Write n in 5-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 16, 21, 10, 11, 12, 17, 22, 15, 16, 17, 18, 23, 20, 21, 22, 23, 24, 25, 30, 55, 80, 105, 30, 31, 56, 81, 106, 55, 56, 61, 86, 111, 80, 81, 86, 91, 116, 105, 106, 111, 116, 121, 50, 55, 60, 85, 110, 55, 56, 61, 86, 111, 60, 61, 62, 87, 112, 85

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..3124

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), this sequence (b=5), A319723 (b=6), A319724 (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165032, A319653.

Table[FromDigits[ReverseSort[IntegerDigits[n, 5]], 5], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 5), , 4), 5); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(5, 100)

n <= a(n) < 5n. - Charles R Greathouse IV, Aug 07 2024

A319723 Write n in 6-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 13, 19, 25, 31, 12, 13, 14, 20, 26, 32, 18, 19, 20, 21, 27, 33, 24, 25, 26, 27, 28, 34, 30, 31, 32, 33, 34, 35, 36, 42, 78, 114, 150, 186, 42, 43, 79, 115, 151, 187, 78, 79, 85, 121, 157, 193, 114, 115, 121, 127, 163, 199, 150, 151, 157, 163

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..7775

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), A319722 (b=5), this sequence (b=6), A319724 (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165051, A319654.

Table[FromDigits[ReverseSort[IntegerDigits[n, 6]], 6], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 6), , 4), 6); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(6, 100)

n <= a(n) < 6n. - Charles R Greathouse IV, Aug 07 2024

A319651 Largest number having in its ternary representation the same number of 0's, 1's and 2's as n.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 7, 8, 9, 12, 21, 12, 13, 22, 21, 22, 25, 18, 21, 24, 21, 22, 25, 24, 25, 26, 27, 36, 63, 36, 39, 66, 63, 66, 75, 36, 39, 66, 39, 40, 67, 66, 67, 76, 63, 66, 75, 66, 67, 76, 75, 76, 79, 54, 63, 72, 63, 66, 75, 72, 75, 78, 63, 66, 75, 66, 67, 76, 75, 76

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..6560

Base b: A073138 (b=2), this sequence (b=3), A319720 (b=4), A319722 (b=5), A319723 (b=6), A319724 (b=7), this sequence (b=8), A319726 (b=9), A004186 (b=10).

Cf. A038574.

Table[FromDigits[ReverseSort[IntegerDigits[n, 3]], 3], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 3),,4), 3); \\ Michel Marcus, Sep 25 2018

from gmpy2 import digits
def A319651(n):
    return int(''.join(sorted(digits(n,3),reverse=True)),3) # Chai Wah Wu, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(3, 100)

n <= a(n) < 3n. - Charles R Greathouse IV, Aug 07 2024

cp's OEIS Frontend

A319785 a(n) = A073138(n) + A038573(n).

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A319650 a(n) = A073138(n) - n.

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A380544 Numbers of the form A073138(k) XOR A038573(k).

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A004086 Read n backwards (referred to as R(n) in many sequences).

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A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

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Magma

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A038573 a(n) = 2^A000120(n) - 1.

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A256999 a(n)=n for n <= 1; for n >= 2, a(n) = largest number that can be obtained by rotating non-msb bits of binary expansion of n (with A080541 or A080542).