A003987 -id:A003987 - cp's OEIS frontend

A285117 Triangle read by rows: T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) XOR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and XOR is bitwise-XOR (A003987).

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 2, 0, 2, 1, 1, 5, 2, 2, 5, 1, 1, 4, 15, 0, 15, 4, 1, 1, 7, 9, 27, 27, 9, 7, 1, 1, 6, 18, 54, 0, 54, 18, 6, 1, 1, 9, 20, 36, 126, 126, 36, 20, 9, 1, 1, 8, 45, 112, 42, 0, 42, 112, 45, 8, 1, 1, 11, 39, 85, 170, 46, 46, 170, 85, 39, 11, 1, 1, 10, 60, 146, 495, 132, 0, 132, 495, 146, 60, 10, 1

Offset: 0

Antti Karttunen, Apr 16 2017

			Rows 0 - 12 of the triangle:
  1,
  1, 1,
  1, 0, 1,
  1, 3, 3, 1,
  1, 2, 0, 2, 1,
  1, 5, 2, 2, 5, 1,
  1, 4, 15, 0, 15, 4, 1,
  1, 7, 9, 27, 27, 9, 7, 1,
  1, 6, 18, 54, 0, 54, 18, 6, 1,
  1, 9, 20, 36, 126, 126, 36, 20, 9, 1,
  1, 8, 45, 112, 42, 0, 42, 112, 45, 8, 1,
  1, 11, 39, 85, 170, 46, 46, 170, 85, 39, 11, 1,
  1, 10, 60, 146, 495, 132, 0, 132, 495, 146, 60, 10, 1

Cf. A003987, A007318, A285116, A285118.

Cf. A285114 (row sums).

T[n_, k_]:= If[n==0 || n==k, 1, BitXor[Binomial[n - 1, k - 1], Binomial[n - 1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 16 2017 *)

T(n, k) = if (n==0 || n==k, 1, bitxor(binomial(n - 1, k - 1), binomial(n - 1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 16 2017

(define (A285117 n) (A285117tr (A003056 n) (A002262 n)))
(define (A285117tr n k) (cond ((zero? k) 1) ((= k n) 1) (else (A003987tr (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A003987bi implements bitwise-XOR (A003987) and A007318tr gives the binomial coefficients (A007318).

T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) XOR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and XOR is bitwise-XOR (A003987).
T(n,k) = A285116(n,k) - A285118(n,k).
C(n,k) = T(n,k) + 2*A285118(n,k). [Where C(n,k) = A007318.]

A286155 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n XOR k,k), else A(n,k) = T(n,n XOR k), where T(n,k) is sequence A000027 considered as a two-dimensional table and XOR is bitwise-xor (A003987).

Original entry on oeis.org

-1, 4, 6, 2, -2, 3, 11, 3, 2, 15, 7, 23, -3, 27, 10, 22, 30, 39, 43, 35, 28, 16, 12, 31, -4, 34, 14, 21, 37, 17, 24, 10, 7, 26, 20, 45, 29, 57, 18, 14, -5, 12, 19, 65, 36, 56, 68, 81, 19, 26, 24, 18, 89, 77, 66, 46, 38, 69, 109, 20, -6, 17, 117, 76, 44, 55, 79, 47, 58, 124, 141, 21, 16, 149, 133, 64, 54, 91, 67, 107, 48, 140, 125, 177, -7, 185, 132, 150, 53, 119

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 1 .. 12 x 1 .. 12 corner of the array:
  -1,   4,   2,  11,   7,  22,  16,  37,  29,  56,  46,  79
   6,  -2,   3,  23,  30,  12,  17,  57,  68,  38,  47, 107
   3,   2,  -3,  39,  31,  24,  18,  81,  69,  58,  48, 139
  15,  27,  43,  -4,  10,  14,  19, 109, 124, 140, 157,  59
  10,  35,  34,   7,  -5,  26,  20, 141, 125, 176, 158,  83
  28,  14,  26,  12,  24,  -6,  21, 177, 196, 142, 159, 111
  21,  20,  19,  18,  17,  16,  -7, 217, 197, 178, 160, 143
  45,  65,  89, 117, 149, 185, 225,  -8,  36,  44,  53,  63
  36,  77,  76, 133, 132, 205, 204,  29,  -9,  64,  54,  87
  66,  44,  64, 150, 186, 148, 184,  38,  58, -10,  55, 115
  55,  54,  53, 168, 167, 166, 165,  48,  47,  46, -11, 147
  91, 119, 151,  63,  87, 115, 147,  59,  83, 111, 143, -12

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
MathWorld, Pairing Function

Cf. A003987, A091255.

Cf. also arrays A285732, A286151, A286153.

(define (A286155 n) (A286155bi (A002260 n) (A004736 n)))
(define (A286155bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (A003987bi row col) col)) (else (A000027bi row (A003987bi col row))))) ;; Where A003987bi implements bitwise-xor (A003987).
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

If n = k, A(n,k) = -n, if n > k, A(n,k) = T(A003987(n,k),k), otherwise [when n < k], A(n,k) = T(n,A003987(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

A378988 a(n) = 2*n XOR 1+sigma(n), where XOR is bitwise-xor, A003987.

Original entry on oeis.org

0, 0, 3, 0, 13, 1, 7, 0, 28, 7, 27, 5, 21, 5, 7, 0, 49, 12, 51, 3, 11, 9, 55, 13, 18, 31, 31, 1, 37, 117, 31, 0, 115, 115, 119, 20, 109, 113, 119, 11, 121, 53, 123, 13, 21, 21, 111, 29, 88, 58, 47, 11, 93, 21, 39, 9, 35, 47, 75, 209, 69, 29, 23, 0, 215, 21, 195, 247, 235, 29, 199, 84, 217, 231, 235, 21, 251, 53, 207

Offset: 1

Antti Karttunen, Dec 16 2024

For any hypothetical quasiperfect number q (for which sigma(q) = 2*q+1, see A336701), a(q) would be equal to 2*q XOR 2*q+2 = 2*(q XOR q+1) = 2*A038712(1+q) = A100892(1+q).

See also A000079 and A235796 concerning the "almost perfect" or "least deficient" numbers that give positions of 0's here.

Cf. A000079 (conjectured to give positions of all 0's), A000396 (positions of 1's), A000203, A003987, A028982 (positions of even terms), A028983 (of odd terms), A038712, A100892, A318467, A336701, A378998, A379009 [= a(n^2)].

Cf. also A235796, A294898, A294899, A318457.

Array[BitXor[2*#, DivisorSigma[1, #] + 1] &, 100] (* Paolo Xausa, Dec 16 2024 *)

PARI
```
A378988(n) = bitxor(n+n,1+sigma(n));
```

For all n in A028983, a(n) = 2n+1 XOR sigma(n) = 1+A318467(n).

A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 0, 0, 6, 0, 14, 14, 14, 0, 30, 30, 30, 30, 30, 18, 16, 0, 62, 62, 62, 62, 62, 50, 48, 62, 62, 34, 32, 34, 32, 44, 44, 0, 126, 126, 126, 126, 126, 114, 112, 126, 126, 98, 96, 98, 96, 108, 108, 126, 126, 66, 64, 66, 64, 76, 76, 66, 64, 92, 92, 92, 92, 92, 82, 0, 254, 254, 254, 254, 254, 242, 240, 254, 254, 226, 224, 226, 224, 236, 236, 254, 254, 194, 192, 194

Offset: 0

Antti Karttunen, Mar 20 2017

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

Cf. A003987, A005187, A006068, A283997.

Table[BitXor[Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]], 2 n - DigitCount[2 n, 2, 1]], {n, 0, 84}] (* Michael De Vlieger, Mar 20 2017, after Jan Mangaldan at A006068 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
a(n) = if(n<2, n, 2*a(floor(n/2)) + (n%2 + a(floor(n/2))%2)%2);
for(n=0, 110, print1(bitxor(A(n),a(n)),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
def a(n): return n if n<2 else 2*a(n//2) + (n%2 + a(n//2)%2)%2
print([A(n)^a(n) for n in range(111)]) # Indranil Ghosh, Mar 25 2017

(define (A283999 n) (A003987bi (A005187 n) (A006068 n))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

a(n) = A006068(2*n) XOR A283997(2*n).

A318460 a(n) = Sum_{d|n, d < n/d} (d XOR n/d), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 3, 2, 5, 4, 8, 6, 15, 8, 18, 10, 24, 12, 20, 20, 27, 16, 35, 18, 30, 24, 32, 22, 52, 24, 42, 36, 44, 28, 56, 30, 63, 40, 54, 36, 81, 36, 56, 52, 90, 40, 80, 42, 80, 68, 68, 46, 116, 48, 93, 68, 86, 52, 112, 68, 112, 72, 90, 58, 144, 60, 92, 98, 119, 72, 136, 66, 122, 88, 128, 70, 171, 72, 114, 110, 136, 88, 152, 78

Offset: 1

Antti Karttunen, Aug 28 2018

Cf. A003987, A037213, A318461, A318462, A318463.

A318460(n) = { my(xors=0); fordiv(n,d, if(d<(n/d), xors += bitxor(d,n/d))); (xors); };

a(n) = (1/2) * Sum_{d|n} (d XOR n/d).

a(n) = A318462(n) - A037213(n).

A325317 a(n) = A048250(n) XOR A162296(n), where XOR is the bitwise-XOR, A003987.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 23, 20, 10, 32, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 48, 54, 48, 67, 38, 60, 56, 90, 42, 96, 44, 20, 46, 72, 48, 124, 57, 89, 72, 18, 54, 96, 72, 120, 80, 90, 60, 40, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 187, 74, 114, 124, 108, 96, 168, 80

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000203, A003987, A048250, A162296, A325316, A325318.

Array[BitXor @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325317(n) = bitxor(A048250(n),A162296(n));

a(n) = A003987(A048250(n), A162296(n)).

a(n) = A000203(n) - 2*A325318(n) = A325316(n) - A325318(n).

A355340 a(0) = 0; for n >= 1, a(n) = a(n-1) XOR A001511(n), where XOR denotes bitwise exclusive-or (A003987) and A001511 is the binary ruler function.

Original entry on oeis.org

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

Offset: 0

Peter Munn, Jun 29 2022

Related to the Thue-Morse sequence, A010060, which gives the rightmost binary bit of each term. The next bit is given by the closely related A269723.
If we replace A001511(n) in the definition by A006519(n) = 2^(A001511(n)-1) we get Gray code (A003188).
Interesting symmetries of the sequence seem more apparent with the terms aligned in suitable periods, such as the arrangement in the example section.

			Initial terms arranged in periods of 16, with deliberate periodic spacing:
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
.
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
...
Note that when the arrangement is partitioned regularly into 2 X 2, 4 X 4 or 8 X 8 squares, the terms on any diagonal of a square share the same value. Note also the symmetry of the terms on the squares' circumferences.

Paolo Xausa, Table of n, a(n) for n = 0..16383
Rémy Sigrist, Colored representation of the first 2^15 terms as 128 rows of 256 terms (the color is function of a(x + 256*y), x = 0..255, y = 0..127)
Index entries for sequences related to binary expansion of n

Comparable sequences: A010060, A261283, A269723.
Cf. A001511, A003188, A003987, A006519.
Positions of: odd numbers: A000069, even numbers: A001969, previously unseen numbers: A253317 (apparently).

Block[{k = 0}, NestList[BitXor[#, IntegerExponent[k += 2, 2]] &, 0, 100]] (* Paolo Xausa, May 29 2024 *)

A010060(n) = a(n) mod 2.

A269723(n) = floor(a(n)/2) mod 2.

A283979 a(n) = (n XOR A264977(n))/4, where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 3, 0, 3, 0, 0, 0, 7, 6, 5, 4, 4, 6, 7, 0, 5, 6, 5, 0, 7, 0, 0, 0, 15, 14, 9, 12, 10, 10, 9, 8, 10, 8, 8, 12, 10, 14, 15, 0, 9, 10, 15, 12, 14, 10, 9, 0, 9, 14, 9, 0, 15, 0, 0, 0, 31, 30, 17, 28, 22, 18, 21, 24, 20, 20, 18, 20, 16, 18, 17, 16, 22, 20, 22, 16, 21, 16, 16, 24, 16, 20, 18, 28, 22, 30, 31, 0, 17, 18, 27, 20, 28, 30

Offset: 0

Antti Karttunen, Mar 25 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003987, A264977, A265397, A284013.

A264977[n_]:= If[n<2, n, If[EvenQ[n], 2 A264977[n/2], BitXor[A264977[(n - 1)/2], A264977[(n + 1)/2]]]]; Table[BitXor[n, A264977[n]]/4, {n, 0, 100}] (* Indranil Ghosh, Mar 28 2017 *)

a(n) = if(n<2, n, if(n%2, bitxor(a((n - 1)/2), a((n + 1)/2)), 2*a(n/2)));
for(n=0, 150, print1(bitxor(n, a(n))/4,", ")) \\ Indranil Ghosh, Mar 28 2017

(define (A283979 n) (/ (A003987bi n (A264977 n)) 4)) ;; Where A003987bi implements bitwise-xor (A003987).

a(n) = (n XOR A264977(n))/4, where XOR is bitwise-xor (A003987).

A329331 Binary operation over the nonnegative integers, distributive over A003987(.,.), such that A(2^i, 2^j) = 2^A054237(i,j). Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 16, 9, 16, 5, 0, 0, 6, 18, 20, 20, 18, 6, 0, 0, 7, 24, 23, 32, 23, 24, 7, 0, 0, 8, 26, 30, 36, 36, 30, 26, 8, 0, 0, 9, 64, 29, 48, 33, 48, 29, 64, 9, 0, 0, 10, 66, 72, 52, 54, 54, 52, 72, 66, 10, 0, 0, 11, 72, 75, 128, 51, 40, 51, 128, 75, 72, 11, 0

Offset: 0

Peter Munn, Nov 10 2019

This sequence defines a multiplication operation that goes with bitwise exclusive-or (A003987) as addition operation to form a ring over the nonnegative integers. It is isomorphic to the polynomial ring GF(2)[x,y], as is the ring defined in A329329.
The ring defined by A329329 is unusual in that it has A059897(.,.) as its addition operation, given that A059897 has more similarities to integer multiplication. A003987, which is isomorphic to A059897 as a binary operation, seems a more standard choice for an addition operator.
However, as explained in A329329, A059897 has a natural choice for mapping a generating set to the 2-dimensions (x and y) of the generating set for the additive group of GF(2)[x,y]. Instead, A003987 needs a pairing function to map its most natural generating set {2^k: k >= 0} onto {x^i * y^j: i >= 0, j >= 0}.
The choice made here was to map 2^k onto the 2 dimensions of x^i * y^j, by proceeding through x and y dimensions as when reading an array by antidiagonals. 2^0 = 1 is mapped to (x^0 * y*0) = 1, 2^1 = 2 is mapped to (x^1 * y^0) = x, 2^2 = 4 to (x^0 * y^1) = y, 8 to (x^2 * y^0) = x^2, and so on, 16 mapped to xy, 32 to y^2, 64 to x^3, etc. With this mapping, it can be shown that the result of the multiplying the polynomial images of 2^i and 2^j is the image of 2^A054237(i,j).

			Square array A(n,k) begins:
  n\k |   0     1     2     3     4     5     6     7     8     9    10
  ----+----------------------------------------------------------------
    0 |   0     0     0     0     0     0     0     0     0     0     0
    1 |   0     1     2     3     4     5     6     7     8     9    10
    2 |   0     2     8    10    16    18    24    26    64    66    72
    3 |   0     3    10     9    20    23    30    29    72    75    66
    4 |   0     4    16    20    32    36    48    52   128   132   144
    5 |   0     5    18    23    36    33    54    51   136   141   154
    6 |   0     6    24    30    48    54    40    46   192   198   216
    7 |   0     7    26    29    52    51    46    41   200   207   210
    8 |   0     8    64    72   128   136   192   200  1024  1032  1088
    9 |   0     9    66    75   132   141   198   207  1032  1025  1098
   10 |   0    10    72    66   144   154   216   210  1088  1098  1032

Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Pairing Function
Eric Weisstein's World of Mathematics, Ring
Wikipedia, Generating set of a group
Wikipedia, Polynomial ring

Cf. A003987, A054237, A059897, A329329.

A(2^i, 2^j) = 2^A054237(i,j).
A(A003987(n,m), k) = A003987(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A003987(A(n,m), A(n,k)).
Derived formulas:(Start)
A(n,k) = A(k,n).
A(n,0) = A(0,k) = 0.
A(n,1) = A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
(End)

A375551 a(n) = Sum_{k=0..n} k XOR n-k, where XOR is the bitwise exclusive disjunction. Row sums of A003987.

Original entry on oeis.org

0, 2, 4, 12, 12, 22, 32, 56, 48, 58, 68, 100, 108, 142, 176, 240, 208, 210, 212, 252, 252, 294, 336, 424, 416, 458, 500, 596, 636, 734, 832, 992, 896, 866, 836, 876, 844, 886, 928, 1048, 1008, 1050, 1092, 1220, 1260, 1390, 1520, 1744, 1680, 1714, 1748, 1884, 1916

Offset: 0

Peter Luschny, Sep 27 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A003986, A003987, A006582, A020522, A051933, A099027, A224915, A376585.

XOR := (n, k) -> Bits:-Xor(n, k):
a := n -> local k; add(XOR(k, n-k), k=0..n):
seq(a(n), n = 0..52);

(* Using definition *)
Table[Sum[BitXor[n - k, k], {k, 0, n}], {n, 0, 100}]
(* Using recurrence -- faster *)
a[0] = 0; a[n_] := a[n] = If[OddQ[n], 4*a[(n-1)/2] + n + 1, 2*(a[n/2] + a[n/2-1])];
Table[a[n], {n, 0, 100}] (* Paolo Xausa, Oct 01 2024 *)

a(n) = sum(k=0, n, bitxor(k, n-k)); \\ Michel Marcus, Sep 28 2024

a(n) = 2*A099027(n).
a(n) = 2*n + A006582(n).
a(2^n - 1) = 4^n - 2^n = A020522(n).
a(2^n) = 4^n - 2^n*(n - 1) = 2*A376585(n).
Recurrence: a(0) = 0; a(2*n) = 2*(a(n) + a(n-1)); a(2*n+1) = 2*(2*a(n) + n + 1). - Paolo Xausa, Oct 01 2024, derived from recurrence in A099027.

cp's OEIS Frontend

A285117 Triangle read by rows: T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) XOR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and XOR is bitwise-XOR (A003987).

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A286155 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n XOR k,k), else A(n,k) = T(n,n XOR k), where T(n,k) is sequence A000027 considered as a two-dimensional table and XOR is bitwise-xor (A003987).

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A378988 a(n) = 2*n XOR 1+sigma(n), where XOR is bitwise-xor, A003987.

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A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

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Formula

A318460 a(n) = Sum_{d|n, d < n/d} (d XOR n/d), where XOR is bitwise-xor (A003987).

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PARI

Formula

A325317 a(n) = A048250(n) XOR A162296(n), where XOR is the bitwise-XOR, A003987.

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Mathematica

PARI

Formula

A355340 a(0) = 0; for n >= 1, a(n) = a(n-1) XOR A001511(n), where XOR denotes bitwise exclusive-or (A003987) and A001511 is the binary ruler function.

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Examples

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Mathematica

Formula

A283979 a(n) = (n XOR A264977(n))/4, where XOR is bitwise-xor (A003987).

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