A001737 -id:A001737 - cp's OEIS frontend

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

List of binary numbers. (This comment is to assist people searching for that particular phrase. - N. J. A. Sloane, Apr 08 2016)
Or, numbers that are sums of distinct powers of 10.
Or, numbers having only digits 0 and 1 in their decimal representation.
Complement of A136399; A064770(a(n)) = a(n). - Reinhard Zumkeller, Dec 30 2007
From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 1.
Thus nonnegative integers n in base 10 such that kn can be calculated by normal addition (i.e., n + n + ... + n, with k n's (but not necessarily k + k + ... + k, with n k's)) or multiplication without requiring any carry operations for 0 <= k <= 9. (End)
For n > 1: A257773(a(n)) = 10, numbers that are Belgian-k for k=0..9. - Reinhard Zumkeller, May 08 2015
For any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
N is in this sequence iff A007953(N) = A101337(N). A028897 is a left inverse. - M. F. Hasler, Nov 18 2019
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023

			a(6)=110 because (1/2)*((1-(-1)^6)*10^0 + (1-(-1)^3)*10^1 + (1-(-1)^1)*10^2) = 10 + 100.
G.f. = x + 10*x^2 + 11*x^3 + 100*x^4 + 101*x^5 + 110*x^6 + 111*x^7 + 1000*x^8 + ...
.
  000    The numbers < 2^n can be regarded as vectors with
  001    a fixed length n if padded with zeros on the left
  010    side. This represents the n-fold Cartesian product
  011    over the set {0, 1}. In the example on the left,
  100    n = 3. (See also the second Python program.)
  101    Binary vectors in this format can also be seen as a
  110    representation of the subsets of a set with n elements.
  111    - _Peter Luschny_, Jan 22 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 21.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991, p. 383.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..32768 (first 8192 terms from Franklin T. Adams-Watters)
Heinz Gumin, Herrn von Leibniz' Rechnung mit Null und Eins, Siemens AG, 3. Auflage 1979 -- contains facsimiles of Leibniz's papers from 1679 and 1703.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
G. W. Leibniz, Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 & 1; avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de l'Académie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).
N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.
Index entries for sequences related to binary expansion of n.

The basic sequences concerning the binary expansion of n are this one, A000120 (Hammingweight: sum of bits), A000788 (partial sums of A000120), A000069 (A000120 is odd), A001969 (A000120 is even), A023416 (number of bits 0), A059015 (partial sums). Bisections A099820 and A099821.
Cf. A028897 (convert binary to decimal).
Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.
Cf. A000290, A001737, A007953, A030308, A054055, A064770, A101337, A136399.

a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n',m) = divMod n 2
-- Reinhard Zumkeller, Jan 10 2012

A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009

Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
Table[Sum[ (Floor[( Mod[f/2 ^n, 2])])*(10^n) , {n, 0, Floor[Log[2, f]]}], {f, 1, 100}] (* José de Jesús Camacho Medina, Jul 24 2014 *)
FromDigits/@Tuples[{1,0},6]//Sort (* Harvey P. Dale, Aug 10 2017 *)

{a(n) = subst( Pol( binary(n)), x, 10)}; /* Michael Somos, Jun 07 2002 */

{a(n) = if( n<=0, 0, n%2 + 10*a(n\2))}; /* Michael Somos, Jun 07 2002 */

a(n)=fromdigits(binary(n),10) \\ Charles R Greathouse IV, Apr 08 2015

def a(n): return int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021

from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
# Peter Luschny, Jan 22 2024

a(n) = Sum_{i=0..m} d(i)*10^i, where Sum_{i=0..m} d(i)*2^i is the base 2 representation of n.
a(n) = (1/2)*Sum_{i>=0} (1-(-1)^floor(n/2^i))*10^i. - Benoit Cloitre, Nov 20 2001
a(n) = A097256(n)/9.
a(2n) = 10*a(n), a(2n+1) = a(2n)+1.
G.f.: 1/(1-x) * Sum_{k>=0} 10^k * x^(2^k)/(1+x^(2^k)) - for sequence as decimal integers. - Franklin T. Adams-Watters, Jun 16 2006
a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009
a(n) = Sum_{k>=0} A030308(n,k)*10^k. - Philippe Deléham, Oct 19 2011
For n > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012
a(n) = Sum_{k=0..floor(log_2(n))} floor((Mod(n/2^k, 2)))*(10^k). - José de Jesús Camacho Medina, Jul 24 2014

A094694 Numbers having in binary representation more 1s than their squares.

Original entry on oeis.org

23, 46, 47, 92, 94, 95, 111, 184, 188, 190, 191, 222, 223, 367, 368, 376, 380, 382, 383, 415, 444, 446, 447, 479, 727, 734, 736, 752, 760, 764, 766, 767, 830, 831, 887, 888, 892, 894, 895, 958, 959, 1451, 1454, 1468, 1471, 1472, 1503, 1504, 1520, 1528

Offset: 1

Reinhard Zumkeller, May 20 2004

A000120(a(n)) > A000120(A000290(a(n))).

Carl R. White, Table of n, a(n) for n = 1..10000

Cf. A077436, A094695, A007088, A001737, A064399, A236514, A351149.

Select[Range[1600], DigitCount[#, 2, 1] > DigitCount[#^2, 2, 1] &] (* Harvey P. Dale, Mar 24 2012 *)

is(n)=hammingweight(n)>hammingweight(n^2) \\ Charles R Greathouse IV, Jan 27 2014

print([k for k in range(0, 1530) if bin(k)[2:].count("1") > bin(k**2)[2:].count("1")]) # Karl-Heinz Hofmann, Feb 07 2022

A221643 Numbers k such that k^2 XOR (k+1)^2 is a square, where XOR is the bitwise XOR operator.

Original entry on oeis.org

0, 3, 4, 23, 24, 43, 44, 76, 111, 112, 115, 139, 164, 180, 183, 248, 264, 323, 327, 348, 411, 479, 480, 499, 611, 699, 747, 787, 943, 976, 1072, 1103, 1111, 1176, 1268, 1388, 1447, 1576, 1684, 1851, 1983, 1984, 2008, 2243, 2692, 3271, 3383, 3452, 3464, 3532, 3679, 3804, 3867

Offset: 1

Alex Ratushnyak, Mar 27 2013

Cf. A145827, A001737.

import math
for i in range(1<<16):
    s = (i*i) ^ ((i+1)*(i+1))
    t = int(math.sqrt(s))
    if s == t*t:
        print(str(i), end=',')

A063009 Write n in binary then square as if written in base 10.

Original entry on oeis.org

0, 1, 100, 121, 10000, 10201, 12100, 12321, 1000000, 1002001, 1020100, 1022121, 1210000, 1212201, 1232100, 1234321, 100000000, 100020001, 100200100, 100220121, 102010000, 102030201, 102212100, 102232321, 121000000, 121022001

Offset: 0

Henry Bottomley, Jul 04 2001

The original description was: "Pre-carry binary squares: write n in binary then square as if written in a base large enough to avoid carries". But I changed it, since I prefer to work in base 10. There is no difference until a(1023). - N. J. A. Sloane, May 21 2002

			a(11)=1022121 since 11 written in binary is 1011 and 1011^2 = 1011000 + 0 + 10110 + 1011 = 1022121.
a(1023) = 1111111111^2 = 1234567900987654321.

Harry J. Smith, Table of n, a(n) for n=0,...,1000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Cf. A007088 for binary numbers, A001737 for binary squares (post-carry), A063010 for carryless binary squares.

a(n) = fromdigits(binary(n))^2; \\ Ruud H.G. van Tol, Dec 08 2022

a(2n) = 100*a(n); a(2n+1) = 100*a(n) + 20*A007088(n) + 1.

A165820 a(n) = the smallest positive integer that, when written in binary, contains both binary n and binary n^2 as substrings.

Original entry on oeis.org

1, 4, 19, 16, 89, 100, 113, 64, 593, 1380, 377, 400, 425, 452, 481, 256, 4385, 1298, 723, 10640, 5561, 11748, 8471, 18456, 1649, 1700, 729, 1808, 1865, 1924, 1985, 1024, 33857, 9250, 36041, 75024, 5477, 11558, 40433, 83520, 1681, 87780, 22329, 92048

Offset: 1

Leroy Quet, Sep 28 2009

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A007088, A001737.

Cf. A165819, A165821, A165822, A294977.

f[n_] := Block[{k = 1, is = IntegerString[n, 2], iss = IntegerString[n^2, 2]}, While[ StringPosition[ IntegerString[k, 2], is] == {} || StringPosition[ IntegerString[k, 2], iss] == {}, k++ ]; k]; Array[f, 44] (* Robert G. Wilson v, Oct 02 2009 *)

a(n) = A294977(n, n^2). - Rémy Sigrist, Mar 03 2018

a(7)-a(44) from Robert G. Wilson v, Oct 02 2009

A091092 In binary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 7, 7, 7, 7, 8

Offset: 0

Reinhard Zumkeller, Dec 18 2003

a(n) = A152487(A000290(n),n). - Reinhard Zumkeller, Dec 06 2008

			a(7)=3: 7->'111', 3 x insert a 0 between the last two 1's:
'110001'->49=7^2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Binary
Wikipedia, Levenshtein Distance
Index entries for sequences related to binary expansion of n

Cf. A091093, A091091, A070939, A000290.

a(n) = LevenshteinDistance(A007088(n), A001737(n)).

A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

Original entry on oeis.org

0, 1, 5, 6, 8, 11, 13, 16, 17, 18, 19, 33, 35, 38, 40, 56, 60, 74, 122, 123, 133, 143, 146, 164, 168, 173, 299, 350, 365, 429, 497, 515, 527, 564, 566, 593, 608, 611, 710, 1031, 1050, 1052, 1059, 1088, 1089, 1090, 1092, 1096, 1105, 1287, 1301, 1316, 1322

Offset: 1

Reinhard Zumkeller, Dec 18 2003

A091092(a(n)) = A091093(a(n)).

Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Binary
Eric Weisstein's World of Mathematics, Ternary

Cf. A007088, A001737, A007089, A001738, A000290.

A094695 Numbers having in binary representation fewer ones than in their squares.

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93, 97, 98, 99

Offset: 1

Reinhard Zumkeller, May 20 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A001737, A007088, A077436, A094694, A159918.

Select[Range[100],DigitCount[#,2,1]Harvey P. Dale, May 29 2017 *)

A000120(a(n)) < A000120(A000290(a(n))).

A358701 a(n) is the least number > 1 that needs n toggles in the trailing bits of its binary representation to become a square.

Original entry on oeis.org

4, 5, 7, 14, 79, 831, 6495, 247614, 7361278, 743300286, 121387475838

Offset: 0

Hugo Pfoertner, Dec 16 2022

			a(0) = 4 = 100 in binary, 0 toggled bits needed;
a(1) = 5 = 101_2, 1 toggled bit -> 100_2 = 4;
a(2) = 7 = 111_2, 2 toggled bits -> 100_2 = 4;
a(3) = 14 = 1110_2, 3 toggled bits -> 1001_2 = 9;
a(4) = 79 = 1001111_2, 4 toggled bits -> 1000000_2 = 64;
a(5) = 831 = 1100111111_2, 5 toggled bits -> 1100010000_2 = 784 = 28^2.

Michael S. Branicky, Go program.
Code Golf StackExchange, Toggle some bits and get a square, coding challenge started by user Arnauld, Aug 18 2018.
Delbert L. Johnson, C# program

Cf. A000120, A000290, A001737, A159918.

// see linked program
(C#) // see linked program

a(9) from Michael S. Branicky, Dec 19 2022

a(10) from Delbert L. Johnson, Apr 13 2024

A098800 Length of longest prefix of binary representation of n that is also a suffix of that of n^2.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 4, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 4, 3, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 3, 5, 0, 1, 3, 4, 0, 4, 3, 1, 0, 1, 0, 6, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 5, 0, 1, 0

Offset: 0

Reinhard Zumkeller, Oct 31 2004

Index entries for sequences related to binary expansion of n
Eric Weisstein's World of Mathematics, Binary

Cf. A001737, A007088, A091269.

cp's OEIS Frontend

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

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A094694 Numbers having in binary representation more 1s than their squares.

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A221643 Numbers k such that k^2 XOR (k+1)^2 is a square, where XOR is the bitwise XOR operator.

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A063009 Write n in binary then square as if written in base 10.

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A165820 a(n) = the smallest positive integer that, when written in binary, contains both binary n and binary n^2 as substrings.

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A091092 In binary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

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A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

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A094695 Numbers having in binary representation fewer ones than in their squares.

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