A007608 -id:A007608 - cp's OEIS frontend

A073789 Numbers in base -8.

0, 1, 2, 3, 4, 5, 6, 7, 170, 171, 172, 173, 174, 175, 176, 177, 160, 161, 162, 163, 164, 165, 166, 167, 150, 151, 152, 153, 154, 155, 156, 157, 140, 141, 142, 143, 144, 145, 146, 147, 130, 131, 132, 133, 134, 135, 136, 137, 120, 121, 122, 123, 124, 125, 126

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A007094, A039724, A073785, A007608, A073786, A073787, A073788, A073790 & A039723.

a073789 0 = 0
a073789 n = a073789 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 8) else (q, r)
             where (q, r) = quotRem n (negate 8)
-- Reinhard Zumkeller, Jul 07 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 8], {n, 0, 60}]

def A073789(n):
    s, q = '', n
    while q >= 8 or q < 0:
        q, r = divmod(q, -8)
        if r < 0:
            q += 1
            r += 8
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073790 Numbers in base -9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 180, 181, 182, 183, 184, 185, 186, 187, 188, 170, 171, 172, 173, 174, 175, 176, 177, 178, 160, 161, 162, 163, 164, 165, 166, 167, 168, 150, 151, 152, 153, 154, 155, 156, 157, 158, 140, 141, 142, 143, 144, 145, 146, 147, 148, 130, 131

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007095, A039724, A073785, A007608, A073786, A073787, A073788, A073789 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits@ Rest@ Reverse@ Mod[ NestWhileList[(# - Mod[ #, b])/-b &, i, # != 0 &], b]; Table[ ToNegaBases[n, 9], {n, 0, 60}]

def A073790(n):
    s, q = '', n
    while q >= 9 or q < 0:
        q, r = divmod(q, -9)
        if r < 0:
            q += 1
            r += 9
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A212529 Negative numbers in base -2.

Original entry on oeis.org

11, 10, 1101, 1100, 1111, 1110, 1001, 1000, 1011, 1010, 110101, 110100, 110111, 110110, 110001, 110000, 110011, 110010, 111101, 111100, 111111, 111110, 111001, 111000, 111011, 111010, 100101, 100100, 100111, 100110, 100001, 100000, 100011, 100010, 101101, 101100, 101111, 101110, 101001, 101000, 101011, 101010, 11010101

Offset: 1

Joerg Arndt, May 20 2012

The formula a(n) = A039724(-n) is slightly misleading because sequence A039724 isn't defined for n < 0, and none of the terms a(n) is a term of A039724. It can be seen as the definition of the extension of A039724 to negative indices. Also, recursive definitions or implementations of A039724 require that function to be defined for negative arguments, and using a generic formula it will work as expected for -n, n > 0. - M. F. Hasler, Oct 18 2018

Joerg Arndt, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), pp. 58-59.

Cf. A039724 (nonnegative numbers in base -2).
Cf. A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4).
Cf. A005352.

a212529 = a039724 . negate  -- Reinhard Zumkeller, Feb 05 2014

a:= proc(n) local d, i, l, m;
      m:= n; l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 2, 'm');
        if d=1 and irem(i, 2)=0 then m:= m+1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2012

negabin[n_] := negabin[n] = If[n == 0, 0, negabin[Quotient[n - 1, -2]]*10 + Mod[n, 2]]; a[n_] := negabin[-n]; Array[a, 50] (* Amiram Eldar, Jul 23 2023 *)

A212529(n)=A039724(-n) \\ M. F. Hasler, Oct 16 2018

def A212529(n):
    s, q = '', -n
    while q >= 2 or q < 0:
        q, r = divmod(q, -2)
        if r < 0:
            q += 1
            r += 2
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

a(n) = A039724(-n). - Reinhard Zumkeller, Feb 05 2014

A271472 Binary representation of n in base i-1.

Original entry on oeis.org

0, 1, 1100, 1101, 111010000, 111010001, 111011100, 111011101, 111000000, 111000001, 111001100, 111001101, 100010000, 100010001, 100011100, 100011101, 100000000, 100000001, 100001100, 100001101, 110011010000, 110011010001, 110011011100, 110011011101, 110011000000, 110011000001

Offset: 0

N. J. A. Sloane, Apr 08 2016

This is A066321 converted from base 10 to base 2.
Every Gaussian integer r+s*i (r, s ordinary integers) has a unique representation as a sum of powers of t = i-1. For example 3 = 1+b^2+b^3, that is, "1101" in binary, which explains a(3) = 1101. See A066321 for further information.
From Jianing Song, Jan 22 2023: (Start)
Also binary representation of n in base -1-i.
Write out n in base -4 (A007608), then change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172. (See also exercise 16, p. 177; answer, p. 494.)
W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
Wikipedia, Complex-base system

Cf. A066321.

a(n) = my(v = [n,0], x=0, digit=0, a, b); while(v!=[0,0], a=v[1]; b=v[2]; v[1]=-2*(a\2)+b; v[2]=-(a\2); x+=(a%2)*10^digit; digit++); x \\ Jianing Song, Jan 22 2023; [a,b] represents the number a + b*(-1+i)

from gmpy2 import c_divmod
u = ('0000','1000','0011','1011')
def A271472(n):
    if n == 0:
        return 0
    else:
        s, q = '', n
        while q:
            q, r = c_divmod(q, -4)
            s += u[r]
        return int(s[::-1]) # Chai Wah Wu, Apr 09 2016

A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, -4, -3, -2, -1, -8, -7, -6, -5, -12, -11, -10, -9, 16, 17, 18, 19, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 32, 33, 34, 35, 28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 48, 49, 50, 51, 44, 45, 46, 47, 40, 41, 42, 43, 36, 37, 38, 39, -64, -63, -62, -61, -68, -67, -66, -65, -72, -71, -70, -69

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 4 representation for n converted from base -4 to base 10.

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], 2020-2021.

Cf. A007090, A007608, A053985, A065369, A073792, A073793, A073794, A073795, A073796 & A073835.

Cf. A066321, A320283.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 4]], {n, 1, 80}]; b

a(n) = subst(Pol(digits(n,4)), x, -4); \\ Michel Marcus, Jan 30 2019

a(4*k+m) = -4*a(k)+m for 0 <= m < 4. - Chai Wah Wu, Jan 16 2020

A066323 Number of one bits in binary representation of base i-1 expansion of n (where i = sqrt(-1)).

Original entry on oeis.org

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

Offset: 0

Marc LeBrun, Dec 14 2001

First differences are usually +1, occasionally -4 (because in base i-1 [3]+[7]=(+i)+(-i)=0) hence often a(i+j)=a(i)+a(j). Differences terms given here are period-16, but for full sequence is actually period-256 at least.

a(n) is the sum of the digits of n when written in base -4 using digits 0 to 3 (A007608). This is since in Penney's digit substitution for A066321, the base -4 digits 0 to 3 become bit strings of exactly 0 to 3 many 1-bits each respectively. - Kevin Ryde, Sep 09 2019

			A066321(4) = 464 = 111010000 (binary) so a(4) = 4.  Or A007608(4) == 130 in base -4 and sum of digits is a(4) = 1+3+0 = 4.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172. (Also exercise 16, p. 177, answer, p. 494.)

David A. Corneth, Table of n, a(n) for n = 0..9999
Walter Penney, A Binary System for Complex Numbers, NSA Technical Journal, 1965.
Walter Penney, A Binary System for Complex Numbers, Journal of the Association for Computing Machinery (JACM), volume 12, number 2, April 1965, pages 247-248.

Cf. A066321, A000120, A007608, A342728.

a[n_] := Plus @@ Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]; Array[a, 100, 0] (* Amiram Eldar, Mar 22 2021 *)

a(n) = my(ret=0); while(n, ret+=n%4; n\=-4); ret; \\ Kevin Ryde, Sep 09 2019

a(n) = A000120(A066321(n)) = A007953(A007608(n)). - Kevin Ryde, Sep 09 2019

A212526 Negative integers in base -4.

Original entry on oeis.org

13, 12, 11, 10, 23, 22, 21, 20, 33, 32, 31, 30, 1303, 1302, 1301, 1300, 1313, 1312, 1311, 1310, 1323, 1322, 1321, 1320, 1333, 1332, 1331, 1330, 1203, 1202, 1201, 1200, 1213, 1212, 1211, 1210, 1223, 1222, 1221, 1220

Offset: 1

Joerg Arndt, May 20 2012

Interleaving with zeros gives A212542 (base 2i representation of negative integers).

More precisely, a(n) is the representation of -n in base -4. - M. F. Hasler, May 21 2012

			a(13)=1303:  1*(-4)^3 + 3*(-4)^2 + 0*(-4)^1 + 3*(-4)^0 = -64 + 48 +3 = -13.

Joerg Arndt, Table of n, a(n) for n = 1..1000

Cf. A007608 (Nonnegative integers in base -4).

a:= proc(n) local d, i, l, m;
      m:= n;
      l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 4, 'm');
        if irem (i, 2)=0 and d>0 then d:= 4-d; m:= m+1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 20 2012

A212526(n,s="")={n=-n;until(!n\=-4,s=Str(n%-4,s));eval(s)}  \\ M. F. Hasler, May 21 2012

def A212526(n):
    s, q = '', -n
    while q >= 4 or q < 0:
        q, r = divmod(q, -4)
        if r < 0:
            q += 1
            r += 4
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 10 2016

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 24, 26, 39, 41, 43, 56, 58, 60, 73, 75, 77, 90, 92, 94, 107, 109, 111, 136, 138, 140, 153, 155, 157, 170, 172, 174, 199, 201, 203, 216, 218, 220, 233, 235, 237, 262, 264, 266, 279, 281, 283, 296, 298, 300, 313, 315, 317, 330, 332, 334, 347, 349

Offset: 1

Amiram Eldar, Mar 19 2021

Equivalently, self numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
Analogous to self numbers (A003052) using base i-1 representation (A271472) instead of decimal expansion.
The number of terms not exceeding 10^k, for k=1,2,..., is 5, 20, 155, 1507, 15008, 150007, 1500014, 15000011. Is the asymptotic density of this sequence exactly 3/20?

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Cf. A007608, A066323, A066321, A271472, A342725, A342726, A342727, A342728.

Similar sequences: A003052 (decimal), A010061 (binary), A010064 (base 4), A010067 (base 6), A010070 (base 8), A339211 (Zeckendorf), A339212 (dual Zeckendorf), A339213 (base phi), A339214 (factorial base), A339215 (primorial base).

s[n_] := Module[{v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}}, Plus @@ Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; f[n_] := n + s[n]; m = 1000; Complement[Range[m], Select[Union@Array[f, m], # <= m &]]

A342728 a(n) is the least number k such that A066323(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 23, 39, 55, 71, 87, 103, 359, 615, 871, 1127, 1383, 1639, 5735, 9831, 13927, 18023, 22119, 26215, 91751, 157287, 222823, 288359, 353895, 419431, 1468007, 2516583, 3565159, 4613735, 5662311, 6710887, 23488103, 40265319, 57042535, 73819751

Offset: 0

Amiram Eldar, Mar 19 2021

a(n) is the least number k whose sum of digits in base i-1 (or in base -4) is n.

Amiram Eldar, Table of n, a(n) for n = 0..1000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,16,-16).

Cf. A007608, A066321, A066323, A271472, A342725, A342726, A342727, A342729.

Join[{0}, LinearRecurrence[{1,0,0,0,0,16,-16}, Range[7], 50]]

a(n) = n for n <= 7, and a(n) = a(n-1) + 16*a(n-6) - 16*a(n-7) for n > 7.
G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 - 15*x^6)/(1 - x - 16*x^6 + 16*x^7). - Stefano Spezia, Mar 20 2021
From Greg Dresden, Jun 21 2021: (Start)
a(3*n+1) = (24 + (4^n)*(25 -  9*(-1)^n))/40.
a(3*n+2) = (24 + (4^n)*(50 +  6*(-1)^n))/40.
a(3*n+3) = (24 + (4^n)*(75 + 21*(-1)^n))/40. (End)

A212494 Base 2i representation of nonnegative integers.

Original entry on oeis.org

0, 1, 2, 3, 10300, 10301, 10302, 10303, 10200, 10201, 10202, 10203, 10100, 10101, 10102, 10103, 10000, 10001, 10002, 10003, 20300, 20301, 20302, 20303, 20200, 20201, 20202, 20203, 20100, 20101, 20102, 20103

Offset: 0

Daniel Forgues, May 18 2012

The use of negabinary dispenses with the need for sign bits and for keeping track of signed and unsigned data types. Similarly, the use of base 2i, or quater-imaginary, dispenses with the need to represent the real and imaginary parts of a complex number separately. (The term "quater-imaginary" appears in Knuth's landmark book on computer programming).
Quater-imaginary, based on the powers of 2i (twice the imaginary unit), uses the digits 0, 1, 2, 3. For purely real positive integers, the quater-imaginary representation is the same as negaquartal (base -4) except that 0's are "riffled" in, corresponding to the odd-indexed powers of 2i which are purely imaginary numbers. Therefore, to obtain the base 2i representations of positive real numbers, the algorithm for base -4 representations can be employed with only a small adjustment.
To obtain the base 2i representation of a complex number a+bi, do as above for the real part, then again for the real part of 2i*(a+bi) = -2b+2ai, giving the digits corresponding to the odd-indexed powers of 2i.
Omitting digits for odd powers of 2i (all 0's for the imaginary parts) (e.g. 20300 --> 230) gives A007608 (nonnegative integers in base -4).

			a(5) = 10301 because 5 = 1*(2i)^4+3*(2i)^2+1*(2i)^0 = 1*16+3*(-4)+1*1

Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (1981): 189

Joerg Arndt, Table of n, a(n) for n = 0..1000
Joerg Arndt, Radix 2i
Donald Knuth, An imaginary number system, Communications of the ACM 3 (4), April 1960, pp. 245-247.
OEIS Wiki, Quater-imaginary base
Wikipedia, Quater-imaginary base

Cf. A212542 (Base 2i representation of negative integers).
Cf. A177505.
Cf. A007608 (Nonnegative integers in base -4).

cp's OEIS Frontend

A073789 Numbers in base -8.

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A073790 Numbers in base -9.

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A212529 Negative numbers in base -2.

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A271472 Binary representation of n in base i-1.

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A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

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A066323 Number of one bits in binary representation of base i-1 expansion of n (where i = sqrt(-1)).

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A212526 Negative integers in base -4.

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