A003987 -id:A003987 - cp's OEIS frontend

A318457 a(n) = n XOR A001065(n), where XOR is bitwise-xor (A003987) and A001065 = sum of proper divisors.

1, 3, 2, 7, 4, 0, 6, 15, 13, 2, 10, 28, 12, 4, 6, 31, 16, 7, 18, 2, 30, 24, 22, 60, 31, 10, 22, 0, 28, 52, 30, 63, 46, 54, 46, 19, 36, 48, 54, 26, 40, 28, 42, 4, 12, 52, 46, 124, 57, 25, 38, 26, 52, 116, 38, 120, 46, 26, 58, 80, 60, 28, 22, 127, 82, 12, 66, 126, 94, 12, 70, 51, 72, 98, 122, 12, 94, 20, 78, 58, 121, 126, 82, 216, 66

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A001065, A003987, A033879, A318456, A318458, A318467.

Cf. A000396 (positions of zeros).

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

PARI
```
A318457(n) = bitxor(n,sigma(n)-n);
```

a(n) = A003987(n, A001065(n)).

a(n) = A000203(n) - 2*A318458(n).

A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 2, 15, 12, 0, 6, 31, 31, 6, 26, 4, 20, 4, 6, 63, 48, 3, 50, 2, 10, 8, 54, 12, 45, 30, 30, 0, 36, 116, 30, 127, 114, 114, 118, 19, 108, 112, 118, 10, 120, 52, 122, 12, 20, 20, 110, 28, 91, 57, 46, 10, 92, 20, 38, 8, 34, 46, 74, 208, 68, 28, 22, 255, 214, 20, 194, 246, 234, 28, 198, 83, 216, 230, 234, 20, 250, 52, 206, 26

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318457, A318458, A318466, A318468.
Cf. A000396 (positions of zeros), A378227 (XOR-Moebius transform), A379234 (fixed points), A379236.
Cf. also A294899, A318457, A378988.

Table[BitXor[2n,DivisorSigma[1,n]],{n,80}] (* Harvey P. Dale, Oct 30 2022 *)

PARI
```
A318467(n) = bitxor(2*n,sigma(n));
```

a(n) = A003987(2*n, A000203(n)).
a(n) = A224880(n) - 2*A318468(n).
a(n) = 2*n XOR (A318457(n)+2*A318458(n)). - Antti Karttunen, Jan 08 2025

A294899 a(n) = A000203(n) XOR A005187(n), where XOR is bitwise-XOR, A003987.

Original entry on oeis.org

0, 0, 0, 0, 14, 6, 3, 0, 29, 0, 31, 10, 25, 1, 2, 0, 50, 5, 55, 12, 7, 13, 50, 18, 48, 27, 26, 13, 40, 112, 25, 0, 112, 116, 115, 29, 97, 117, 114, 20, 101, 49, 126, 1, 24, 16, 105, 34, 102, 60, 42, 7, 80, 16, 33, 21, 62, 42, 77, 220, 75, 23, 16, 0, 212, 18, 199, 248, 231, 25, 194, 77, 197, 227, 238, 25, 246, 48, 201, 36

Offset: 1

Antti Karttunen, Nov 25 2017

nonn

Cf. A000203, A003987, A005187, A294898, A295296 (positions of zeros), A295297 (parity of a(n)).

Cf. also A169813, A279357, A283997.

Array[BitXor[DivisorSigma[1, #], IntegerExponent[(2 #)!, 2]] &, 80] (* Michael De Vlieger, Nov 26 2017 *)

(define (A294899 n) (A003987bi (A000203 n) (A005187 n))) ;; Where A003987bi implements the bitwise-XOR, A003987.

a(n) = A000203(n) XOR A005187(n).

A286151 Square array read by descending antidiagonals: If n > k, A(n,k) = T(n XOR k, k), and otherwise A(n,k) = T(n, n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 11, 13, 9, 10, 7, 5, 8, 14, 15, 22, 8, 7, 26, 20, 21, 16, 38, 9, 42, 19, 27, 28, 37, 47, 58, 62, 52, 43, 35, 36, 29, 23, 48, 14, 51, 25, 34, 44, 45, 56, 30, 39, 19, 16, 41, 33, 64, 54, 55, 46, 80, 31, 25, 20, 23, 32, 88, 53, 65, 66, 79, 93, 108, 32, 41, 39, 31, 116, 102, 89, 77, 78, 67, 57, 94, 140, 33, 27, 30, 148, 101, 63, 76, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   2,   2,  11,   7,  22,  16,  37,  29,  56,  46,  79,  67, 106
   5,  13,   5,   8,  38,  47,  23,  30,  80,  93,  57,  68, 138
   9,   8,   7,   9,  58,  48,  39,  31, 108,  94,  81,  69, 174
  14,  26,  42,  62,  14,  19,  25,  32, 140, 157, 175, 194,  82
  20,  19,  52,  51,  16,  20,  41,  33, 176, 158, 215, 195, 110
  27,  43,  25,  41,  23,  39,  27,  34, 216, 237, 177, 196, 142
  35,  34,  33,  32,  31,  30,  29,  35, 260, 238, 217, 197, 178
  44,  64,  88, 116, 148, 184, 224, 268,  44,  53,  63,  74,  86
  54,  53, 102, 101, 166, 165, 246, 245,  46,  54,  87,  75, 114
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  65,  76, 146
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  77, 182
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  90

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000217 (row 0), A000096 (column 0 and the main diagonal).
Cf. A001477, A003987, A286108, A286109, A286145, A286147, A286150.
Cf. A286153 (same array without row 0 and column 0).

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=If[n>k, T[BitXor[n, k], k], T[n, BitXor[n, k]]]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 20 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k) if n>k else T(n, n^k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286151 n) (A286151bi (A002262 n) (A025581 n)))
(define (A286151bi row col) (define (pairA001477bi a b) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2)) (cond ((> row col) (pairA001477bi (A003987bi row col) col)) (else (pairA001477bi row (A003987bi col row))))) ;; Where A003987bi implements bitwise-xor (A003987).

If n > k, A(n,k) = T(A003987(n,k),k), otherwise A(n,k) = T(n,A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

A275808 a(0) = 0, for n >= 1, a(n) = A275736(n) XOR a(A257684(n)), where XOR is given by A003987.

Original entry on oeis.org

0, 1, 2, 3, 1, 0, 4, 5, 6, 7, 5, 4, 2, 3, 0, 1, 3, 2, 1, 0, 3, 2, 0, 1, 8, 9, 10, 11, 9, 8, 12, 13, 14, 15, 13, 12, 10, 11, 8, 9, 11, 10, 9, 8, 11, 10, 8, 9, 4, 5, 6, 7, 5, 4, 0, 1, 2, 3, 1, 0, 6, 7, 4, 5, 7, 6, 5, 4, 7, 6, 4, 5, 2, 3, 0, 1, 3, 2, 6, 7, 4, 5, 7, 6, 0, 1, 2, 3, 1, 0, 3, 2, 1, 0, 2, 3, 1, 0, 3, 2, 0

Offset: 0

Antti Karttunen, Aug 09 2016

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A003987, A248663, A257684, A275734, A275736.
Cf. A275809 (positions of zeros), A275810 (and their first differences).
Cf. also A275728.

a(0) = 0, for n >= 1, a(n) = A275736(n) XOR a(A257684(n)), where 2-argument function XOR is given by A003987.

a(n) = A248663(A275734(n)).

A283987 a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

1, 0, 3, 3, 2, 1, 1, 2, 5, 7, 6, 7, 7, 6, 7, 5, 4, 1, 3, 4, 11, 13, 2, 5, 5, 2, 13, 11, 4, 3, 1, 4, 7, 3, 12, 13, 15, 12, 13, 9, 8, 3, 5, 8, 9, 11, 14, 11, 11, 14, 11, 9, 8, 5, 3, 8, 9, 13, 12, 15, 13, 12, 3, 7, 6, 1, 13, 14, 11, 7, 4, 9, 11, 4, 25, 21, 22, 27, 7, 14, 13, 5, 24, 27, 29, 24, 31, 23, 20, 29, 31, 20, 23, 25, 2, 9, 9, 2, 25, 23, 20, 31, 29, 20, 23

Offset: 1

Antti Karttunen, Mar 21 2017

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

Odd bisection of A283977.
Cf. A002487, A003987, A283988.
Cf. A283973 (positions where coincides with A007306, or equally, with A283986).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitXor[a[n - 1], a@ n], {n, 120}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
for(n=1, 120, print1(bitxor(A(n - 1), A(n)), ", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283987(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))^sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) if n>1 else 1 # Chai Wah Wu, May 05 2023

(define (A283987 n) (A003987bi (A002487 (- n 1)) (A002487 n))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).
a(n) = A283986(n) - A283988(n).
a(n) = A007306(n) - 2*A283988(n).
a(n) = A283977((2*n)-1).

A286108 Square array read by antidiagonals: A(n,k) = T(2*(n AND k), n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 1, 3, 5, 3, 6, 6, 6, 6, 10, 12, 14, 12, 10, 15, 15, 19, 19, 15, 15, 21, 23, 21, 27, 21, 23, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 38, 40, 38, 44, 38, 40, 38, 36, 45, 45, 49, 49, 53, 53, 49, 49, 45, 45, 55, 57, 55, 61, 63, 65, 63, 61, 55, 57, 55, 66, 66, 66, 66, 74, 74, 74, 74, 66, 66, 66, 66, 78, 80, 82, 80, 78, 88, 90, 88, 78, 80, 82, 80, 78

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,  1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   1,  5,   6,  12,  15,  23,  28,  38,  45,  57,  66,  80,  91
   3,  6,  14,  19,  21,  28,  40,  49,  55,  66,  82,  95, 105
   6, 12,  19,  27,  28,  38,  49,  61,  66,  80,  95, 111, 120
  10, 15,  21,  28,  44,  53,  63,  74,  78,  91, 105, 120, 144
  15, 23,  28,  38,  53,  65,  74,  88,  91, 107, 120, 138, 161
  21, 28,  40,  49,  63,  74,  90, 103, 105, 120, 140, 157, 179
  28, 38,  49,  61,  74,  88, 103, 119, 120, 138, 157, 177, 198
  36, 45,  55,  66,  78,  91, 105, 120, 152, 169, 187, 206, 226
  45, 57,  66,  80,  91, 107, 120, 138, 169, 189, 206, 228, 247
  55, 66,  82,  95, 105, 120, 140, 157, 187, 206, 230, 251, 269
  66, 80,  95, 111, 120, 138, 157, 177, 206, 228, 251, 275, 292
  78, 91, 105, 120, 144, 161, 179, 198, 226, 247, 269, 292, 324

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000217 (row 0 & column 0), A014106 (main diagonal).
Cf. A003056, A003987, A004198.
Cf. also arrays A286098, A286109, A286145, A286147, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[2*BitAnd[n, k], BitXor[n, k]]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 20 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(2*(n&k), n^k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286108 n) (A286108bi (A002262 n) (A025581 n)))
(define (A286108bi row col) (let ((a (* 2 (A004198bi row col))) (b (A003987bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003987bi and A004198bi implement bitwise-xor (A003987) and bitwise-and (A004198).

A(n,k) = T(2*A004198(n,k), A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286109 Square array read by antidiagonals: A(n,k) = T(n XOR k, 2*(n AND k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 2, 2, 5, 3, 5, 9, 9, 9, 9, 14, 12, 10, 12, 14, 20, 20, 16, 16, 20, 20, 27, 25, 27, 21, 27, 25, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 42, 40, 42, 36, 42, 40, 42, 44, 54, 54, 50, 50, 46, 46, 50, 50, 54, 54, 65, 63, 65, 59, 57, 55, 57, 59, 65, 63, 65, 77, 77, 77, 77, 69, 69, 69, 69, 77, 77, 77, 77, 90, 88, 86, 88, 90, 80, 78, 80, 90, 88, 86, 88, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
   2,   3,   9,  12,  20,  25,  35,  42,  54,  63,  77,  88, 104
   5,   9,  10,  16,  27,  35,  40,  50,  65,  77,  86, 100, 119
   9,  12,  16,  21,  35,  42,  50,  59,  77,  88, 100, 113, 135
  14,  20,  27,  35,  36,  46,  57,  69,  90, 104, 119, 135, 144
  20,  25,  35,  42,  46,  55,  69,  80, 104, 117, 135, 150, 162
  27,  35,  40,  50,  57,  69,  78,  92, 119, 135, 148, 166, 181
  35,  42,  50,  59,  69,  80,  92, 105, 135, 150, 166, 183, 201
  44,  54,  65,  77,  90, 104, 119, 135, 136, 154, 173, 193, 214
  54,  63,  77,  88, 104, 117, 135, 150, 154, 171, 193, 212, 236
  65,  77,  86, 100, 119, 135, 148, 166, 173, 193, 210, 232, 259
  77,  88, 100, 113, 135, 150, 166, 183, 193, 212, 232, 253, 283
  90, 104, 119, 135, 144, 162, 181, 201, 214, 236, 259, 283, 300

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000096 (row 0 & column 0), A014105 (main diagonal).
Cf. A003056, A003987, A004198.
Cf. also arrays A286099, A286108, A286145, A286147, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], 2*BitAnd[n, k]]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 20 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, 2*(n&k))
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286109 n) (A286109bi (A002262 n) (A025581 n)))
(define (A286109bi row col) (let ((a (A003987bi row col)) (b (* 2 (A004198bi row col)))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003987bi and A004198bi implement bitwise-xor (A003987) and bitwise-and (A004198).

A(n,k) = T(A003987(n,k), 2*A004198(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286145 Square array read by antidiagonals: A(n,k) = T(n XOR k, k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 4, 2, 12, 1, 5, 24, 18, 13, 9, 40, 17, 3, 8, 14, 60, 50, 11, 7, 26, 20, 84, 49, 61, 6, 42, 19, 27, 112, 98, 85, 73, 62, 52, 43, 35, 144, 97, 59, 72, 10, 51, 25, 34, 44, 180, 162, 83, 71, 22, 16, 41, 33, 64, 54, 220, 161, 181, 70, 38, 15, 23, 32, 88, 53, 65, 264, 242, 221, 201, 58, 48, 39, 31, 116, 102, 89, 77, 312, 241, 179, 200, 222, 47, 21, 30, 148, 101, 63, 76, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   4,  12,  24,  40,  60,  84, 112, 144, 180, 220, 264, 312
   2,   1,  18,  17,  50,  49,  98,  97, 162, 161, 242, 241, 338
   5,  13,   3,  11,  61,  85,  59,  83, 181, 221, 179, 219, 365
   9,   8,   7,   6,  73,  72,  71,  70, 201, 200, 199, 198, 393
  14,  26,  42,  62,  10,  22,  38,  58, 222, 266, 314, 366, 218
  20,  19,  52,  51,  16,  15,  48,  47, 244, 243, 340, 339, 240
  27,  43,  25,  41,  23,  39,  21,  37, 267, 315, 265, 313, 263
  35,  34,  33,  32,  31,  30,  29,  28, 291, 290, 289, 288, 287
  44,  64,  88, 116, 148, 184, 224, 268,  36,  56,  80, 108, 140
  54,  53, 102, 101, 166, 165, 246, 245,  46,  45,  94,  93, 158
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  55,  79, 177
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  66, 197
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  78

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
MathWorld, Pairing Function

Transpose: A286147.
Cf. A046092 (row 0), A000096 (column 0), A000217 (main diagonal).
Cf. A003987, A001477, A286108, A286109, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], k]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286145 n) (A286145bi (A002262 n) (A025581 n)))
(define (A286145bi row col) (let ((a (A003987bi row col)) (b col)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

A(n,k) = T(A003987(n,k), k), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286147 Square array read by antidiagonals: A(n,k) = T(n XOR k, n), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 2, 4, 5, 1, 12, 9, 13, 18, 24, 14, 8, 3, 17, 40, 20, 26, 7, 11, 50, 60, 27, 19, 42, 6, 61, 49, 84, 35, 43, 52, 62, 73, 85, 98, 112, 44, 34, 25, 51, 10, 72, 59, 97, 144, 54, 64, 33, 41, 16, 22, 71, 83, 162, 180, 65, 53, 88, 32, 23, 15, 38, 70, 181, 161, 220, 77, 89, 102, 116, 31, 39, 48, 58, 201, 221, 242, 264, 90, 76, 63, 101, 148, 30, 21, 47, 222, 200, 179, 241, 312

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
    0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
    4,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
   12,  18,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
   24,  17,  11,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
   40,  50,  61,  73,  10,  16,  23,  31, 148, 166, 185, 205,  86
   60,  49,  85,  72,  22,  15,  39,  30, 184, 165, 225, 204, 114
   84,  98,  59,  71,  38,  48,  21,  29, 224, 246, 183, 203, 146
  112,  97,  83,  70,  58,  47,  37,  28, 268, 245, 223, 202, 182
  144, 162, 181, 201, 222, 244, 267, 291,  36,  46,  57,  69,  82
  180, 161, 221, 200, 266, 243, 315, 290,  56,  45,  81,  68, 110
  220, 242, 179, 199, 314, 340, 265, 289,  80,  94,  55,  67, 142
  264, 241, 219, 198, 366, 339, 313, 288, 108,  93,  79,  66, 178
  312, 338, 365, 393, 218, 240, 263, 287, 140, 158, 177, 197,  78

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
MathWorld, Pairing Function

Transpose: A286145.
Cf. A000096 (row 0), A046092 (column 0), A000217 (main diagonal).
Cf. A003987, A001477, A286108, A286109, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], k]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k)
for n in range(21): print([A(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286147 n) (A286147bi (A002262 n) (A025581 n)))
(define (A286147bi row col) (let ((a (A003987bi row col)) (b row)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

A(n,k) = T(A003987(n,k), n), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

cp's OEIS Frontend

A318457 a(n) = n XOR A001065(n), where XOR is bitwise-xor (A003987) and A001065 = sum of proper divisors.

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A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

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A294899 a(n) = A000203(n) XOR A005187(n), where XOR is bitwise-XOR, A003987.

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A286151 Square array read by descending antidiagonals: If n > k, A(n,k) = T(n XOR k, k), and otherwise A(n,k) = T(n, n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

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A275808 a(0) = 0, for n >= 1, a(n) = A275736(n) XOR a(A257684(n)), where XOR is given by A003987.

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A283987 a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).

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A286108 Square array read by antidiagonals: A(n,k) = T(2*(n AND k), n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).

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A286109 Square array read by antidiagonals: A(n,k) = T(n XOR k, 2*(n AND k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).

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