A048724 -id:A048724 - cp's OEIS frontend

A178733 a(n) = n XOR 7n, where XOR is bitwise XOR.

0, 6, 12, 22, 24, 38, 44, 54, 48, 54, 76, 70, 88, 86, 108, 102, 96, 102, 108, 150, 152, 134, 140, 182, 176, 182, 172, 166, 216, 214, 204, 198, 192, 198, 204, 214, 216, 294, 300, 310, 304, 310, 268, 262, 280, 278, 364, 358, 352, 358, 364, 342, 344, 326, 332, 438

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 7 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178735 a(n) = n XOR 9n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 8, 16, 24, 32, 40, 48, 56, 64, 88, 80, 104, 96, 120, 112, 136, 128, 136, 176, 184, 160, 168, 208, 216, 192, 248, 240, 232, 224, 280, 272, 264, 256, 264, 272, 280, 352, 360, 368, 376, 320, 344, 336, 424, 416, 440, 432, 392, 384, 392, 496, 504, 480, 488, 464

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178734, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 9 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178736 a(n) = n XOR 10n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 11, 22, 29, 44, 55, 58, 65, 88, 83, 110, 101, 116, 143, 130, 153, 176, 187, 166, 173, 220, 199, 202, 241, 232, 227, 286, 277, 260, 319, 306, 297, 352, 363, 374, 381, 332, 343, 346, 417, 440, 435, 398, 389, 404, 495, 482, 505, 464, 475, 454, 461, 572, 551

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178734, A178735. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 10 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A270437 Multiplicative with a(p^e) = p^(e XOR 2e), where XOR is bitwise-xor.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 32, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 864, 15625, 17576, 243, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 117649, 125000

Offset: 1

Antti Karttunen, May 27 2016

Multiplicative with a(p^e) = p^A048724(e), where A048724(e) = (e XOR 2e).

Multiples of 8 in the ring defined in A329329. - Peter Munn, Jan 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A020639, A028234, A048724, A067029.
Cf. A262675 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270436 and permutation A273671.
Row 8 and column 8 of A329329.

f[p_, e_] := p^BitXor[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 07 2023 *)

a(1) = 1, for n > 1, a(n) = A020639(n)^A048724(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = 1, A270419(a(n)) = n.
a(n) = A329329(n,8) = A329329(8,n). - Peter Munn, Jan 17 2020

Name changed by Antti Karttunen, Sep 07 2023

A292372 A binary encoding of 2-digits in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        1          2           1
   3        0          3           0
   4        0         10           0
   5        0         11           0
   6        1         12           1
   7        0         13           0
   8        2         20          10
   9        2         21          10
  10        3         22          11
  11        2         23          10
  12        0         30           0
  13        0         31           0
  14        1         32           1
  15        0         33           0
  16        0        100           0
  17        0        101           0
  18        1        102           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004198, A007088, A007090, A048724, A059906, A160382, A292370, A292371, A292373.

Cf. A289814 (analogous sequence for base-3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 2, 1, 0], 2], {n, 0, 120}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==2 else '0' for i in k), 2)
print([a(n) for n in range(121)]) # Indranil Ghosh, Sep 21 2017

def A292372(n): return 0 if (m:=n&~(n<<1)) < 2 else int(bin(m)[-2:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059906(n AND A048724(n)), where AND is a bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160382(n).

A277825 a(n) = A048725(A065621(n)) = A048720(A065621(n),5).

Original entry on oeis.org

5, 10, 27, 20, 57, 54, 39, 40, 125, 114, 99, 108, 65, 78, 95, 80, 245, 250, 235, 228, 201, 198, 215, 216, 141, 130, 147, 156, 177, 190, 175, 160, 485, 490, 507, 500, 473, 470, 455, 456, 413, 402, 387, 396, 417, 430, 447, 432, 277, 282, 267, 260, 297, 294, 311, 312, 365, 354, 371, 380, 337, 350, 335, 320, 965, 970, 987, 980, 1017, 1014, 999, 1000

Offset: 1

Antti Karttunen, Nov 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Column 3 of A277820, Column 5 of A277320.

Cf. A048720, A048724, A048725, A065621, A277823.

def A277825(n): return (m:=n^ (n&~-n)<<1)^m<<2 # Chai Wah Wu, Jun 29 2022

(define (A277825 n) (A048724 (A277823 n)))
(define (A277825 n) (A048720bi (A065621 n) 5)) ;; For A048720bi see the Scheme-program attachment in A048720.

a(n) = A048724(A277823(n)) = A048725(A065621(n)).

a(n) = A048720(A065621(n),5).

A295881 Reversing binary representation of the deficiency of n, A033879(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 14, 1, 13, 2, 26, 12, 28, 4, 14, 1, 16, 5, 50, 6, 26, 8, 62, 20, 55, 26, 22, 0, 44, 20, 38, 1, 50, 22, 62, 53, 100, 16, 62, 30, 104, 20, 122, 4, 28, 52, 118, 36, 121, 11, 38, 14, 84, 20, 110, 24, 98, 42, 74, 80, 76, 44, 62, 1, 118, 20, 194, 26, 122, 12, 206, 85, 200, 98, 42, 28, 74, 20, 214, 46, 121

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

For all n, A010060(a(A005100(n))) = 1 and A010060(a(A023196(n))) = 0. That is, for the deficient numbers a(n) is an odious number (A000069) and for the nondeficient numbers a(n) is an evil number (A001969).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005100, A023196, A033879, A033880, A048724, A065620, A065621, A286449, A295882.

Cf. A000396 (gives the positions of zeros).

(define (A295881 n) (let ((x (A033879 n))) (if (<= x 0) (A048724 (- x)) (A065621 x))))

If A033879(n) <= 0, a(n) = A048724(-A033879(n)), otherwise a(n) = A065621(A033879(n)).

For all n >= 1, A065620(a(n)) = A033879(n).

A104895 a(0)=0; thereafter a(2n) = -2a(n), a(2n+1) = 2a(n) - 1.

Original entry on oeis.org

0, -1, -2, 1, -4, 3, 2, -3, -8, 7, 6, -7, 4, -5, -6, 5, -16, 15, 14, -15, 12, -13, -14, 13, 8, -9, -10, 9, -12, 11, 10, -11, -32, 31, 30, -31, 28, -29, -30, 29, 24, -25, -26, 25, -28, 27, 26, -27, 16, -17, -18, 17, -20, 19, 18, -19, -24, 23, 22, -23, 20, -21, -22, 21, -64, 63, 62, -63, 60, -61, -62, 61, 56, -57, -58, 57, -60, 59

Offset: 0

Philippe Deléham, Apr 24 2005

Columns of table in A104894 written in base 10.

Conjecture: the positions where 0, 1, 2, 3, ... appear are given by A048724; the positions where -1, -2, -3, ... appear are given by A065621.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A048724, A065621, A104894.

The negative of entry A065620.

import Data.List (transpose)
a104895 n = a104895_list !! n
a104895_list = 0 : concat (transpose [map (negate . (+ 1)) zs, tail zs])
               where zs = map (* 2) a104895_list
-- Reinhard Zumkeller, Mar 26 2014

f:=proc(n) option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(2*f(n/2)); else RETURN(-2*f((n-1)/2)-1); fi; end;

a[0] = 0;
a[n_]:= a[n]= If[EvenQ[n], 2 a[n/2], -2 a[(n-1)/2] - 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 03 2018 *)

def a(n):
    if (n==0): return 0
    elif (mod(n,2)==0): return 2*a(n/2)
    else: return -2*a((n-1)/2) - 1
[a(n) for n in (0..100)] # G. C. Greubel, Jun 15 2021

a(0) = 0 and for k>=0, 0<= j <2^k, a(2^k + j) = a(j) + 2^k if a(j)<0, a(2^k + j) = a(j) - 2^k if a(j)>=0.
Sum_{0 <= n <= 2^k - 1} a(n) = - 2^(k-1).
Sum_{0 <= n <= 2^k - 1} |a(n)| = 4^(k-1).
a(n) = -A065620(n). - M. F. Hasler, Apr 16 2018

Corrected by N. J. A. Sloane, Nov 05 2005

Edited by N. J. A. Sloane, Apr 25 2018

A213064 Bitwise AND of 2n with the one's-complement of n.

Original entry on oeis.org

0, 2, 4, 4, 8, 10, 8, 8, 16, 18, 20, 20, 16, 18, 16, 16, 32, 34, 36, 36, 40, 42, 40, 40, 32, 34, 36, 36, 32, 34, 32, 32, 64, 66, 68, 68, 72, 74, 72, 72, 80, 82, 84, 84, 80, 82, 80, 80, 64, 66, 68, 68, 72, 74, 72, 72, 64, 66, 68, 68, 64, 66, 64, 64, 128, 130, 132

Offset: 0

Juli Mallett, Jun 04 2012

In two's-complement binary arithmetic, -n is ~(n - 1). As such, this could be written instead as a(n) = 2n AND -(n + 1). Further, because the least significant bits are never matched both of the operands to the AND, the negative form of n can be used rather than the one's-complement, i.e. a(n) = 2n AND -n.

a(n) has a 1-bit immediately above each run of 1's in n, and everywhere else 0's. Or equivalently, each 01 bit pair in n becomes 10 in a(n) and everywhere else 0's. The most significant 1-bit of n has a 0 above it for this purpose, so is an 01 bit pair. - Kevin Ryde, Jun 04 2020

			For n = 31, 2n is 62, which in binary is 111110, as multiplication by two is the same as shifting the bits of 31 (11111) to the left by one. As the number is one less than a power of two, all of its least significant bits are set. Before the shift, the most significant bit has a value of 16. After the shift, the most significant bit has a value of 32.
The ~n has all bits set but the five least significant, the highest bit set being the power of two above n: .....111111111100000. When these two values are ANDed together, only the 6th bit, that with the value of 32, is common to them, and the result is 32.
From _Kevin Ryde_, Jun 04 2020: (Start)
     n = 1831 = binary  11100100111
  a(n) = 2120 = binary 100001001000   1-bit above each run
(End)

Cf. A048724 (with XOR).

int a(int n) { return ((n + n) & ~n); }

Table[BitAnd[2n, -n], {n, 0, 66}] (* Alonso del Arte, Jun 04 2012 *)

a(n) = bitnegimply(n<<1,n); \\ Kevin Ryde, Jun 04 2020

def A213064(n): return n<<1&~n # Chai Wah Wu, Jun 29 2022

# with bitops
bitAnd(2 * n, bitFlip(n))

a(n) = 2n AND ~n

a(n) = 2*A292272(n). - Kevin Ryde, Jun 04 2020

A268386 a(n) = A193231(A268387(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

A003987, A048720, A059897, A193231, A268385, A268387 are used in definitions of this sequence.

Cf. A000028 (indices of odd numbers), A000379 (indices of even numbers), A268390 (indices of zeros).

f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; {0}~Join~Table[f[BitXor @@ Map[Last, FactorInteger@ n]], {n, 2, 120}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)

a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }
a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
a(n) = a193231(a268387(n)); \\ Michel Marcus, May 09 2020

(define (A268386 n) (A193231 (A268387 n)))

The following two formulas are equivalent because A193231 distributes over bitwise XOR (A003987):
a(n) = A193231(A268387(n)) and
a(n) = A268387(A268385(n)).
a(2^k) = A193231(k). - Peter Munn, May 07 2020
From Peter Munn, Jun 02 2020: (Start)
Alternative definition, for n, k >= 1, where XOR denotes A003987:
a(prime(n)) = 1, where prime(n) = A000040(n);
a(n^2) = a(n) XOR (2 * a(n)) = A048720(a(n), 3);
a(A059897(n, k)) = a(n) XOR a(k).
(End)

cp's OEIS Frontend

A178733 a(n) = n XOR 7n, where XOR is bitwise XOR.

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A178735 a(n) = n XOR 9n, where XOR is bitwise XOR.

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A178736 a(n) = n XOR 10n, where XOR is bitwise XOR.

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A270437 Multiplicative with a(p^e) = p^(e XOR 2e), where XOR is bitwise-xor.

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A292372 A binary encoding of 2-digits in base-4 representation of n.

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A277825 a(n) = A048725(A065621(n)) = A048720(A065621(n),5).

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A295881 Reversing binary representation of the deficiency of n, A033879(n).

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A104895 a(0)=0; thereafter a(2n) = -2*a(n), a(2n+1) = 2*a(n) - 1.

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A104895 a(0)=0; thereafter a(2n) = -2a(n), a(2n+1) = 2a(n) - 1.