A268725 -id:A268725 - cp's OEIS frontend

A268723 Main diagonal of A268725: a(n) = A003188(A006068(n)^2), where A003188 is binary Gray code and A006068 is its inverse.

0, 1, 13, 6, 41, 54, 24, 21, 145, 166, 216, 253, 96, 121, 69, 86, 545, 582, 664, 749, 864, 841, 949, 1014, 384, 433, 477, 486, 793, 278, 344, 357, 2113, 2182, 2328, 2509, 2656, 2793, 2901, 2998, 3456, 3537, 3901, 3366, 3641, 3798, 4056, 3973, 1536, 1633, 1709, 1734, 1801, 1910, 1944, 2037, 3313, 3174, 1112, 1053

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000290, A003188, A006068.

Main diagonal of array A268725.

def A268723(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=k**2)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268723 n) (A003188 (A000290 (A006068 n))))

a(n) = A003188(A000290(A006068(n))).

A341520 Square array A(n,k) = A156552(A005940(1+n)*A005940(1+k)), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 5, 3, 4, 7, 6, 7, 4, 5, 9, 11, 11, 9, 5, 6, 11, 10, 15, 10, 11, 6, 7, 13, 13, 19, 19, 13, 13, 7, 8, 15, 14, 23, 12, 23, 14, 15, 8, 9, 17, 23, 27, 21, 21, 27, 23, 17, 9, 10, 19, 18, 31, 22, 27, 22, 31, 18, 19, 10, 11, 21, 21, 35, 39, 29, 29, 39, 35, 21, 21, 11, 12, 23, 22, 39, 20, 47, 30, 47, 20, 39, 22, 23, 12

Offset: 0

Antti Karttunen, Feb 13 2021

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.
This array defines a binary operation on the nonnegative integers that matches up the zeros in the binary representation of each operand (starting from the right, and including as many leading zeros as necessary) and concatenates the two (possibly null) strings of ones to the right of each matched pair of zeros. See the examples. - Peter Munn, Feb 14 2021.
As such it could be useful for implementing multiplication, say, in Turing machines, with a "tape-like" unary-binary encoding of the prime factorization of n (A156552). However, such representation is not very useful if addition or subtraction is also needed.

			The top left {0..15} X {0..16} corner of the array:
   0,  1,  2,  3,  4,  5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15,
   1,  3,  5,  7,  9, 11,  13,  15,  17,  19,  21,  23,  25,  27,  29,  31,
   2,  5,  6, 11, 10, 13,  14,  23,  18,  21,  22,  27,  26,  29,  30,  47,
   3,  7, 11, 15, 19, 23,  27,  31,  35,  39,  43,  47,  51,  55,  59,  63,
   4,  9, 10, 19, 12, 21,  22,  39,  20,  25,  26,  43,  28,  45,  46,  79,
   5, 11, 13, 23, 21, 27,  29,  47,  37,  43,  45,  55,  53,  59,  61,  95,
   6, 13, 14, 27, 22, 29,  30,  55,  38,  45,  46,  59,  54,  61,  62, 111,
   7, 15, 23, 31, 39, 47,  55,  63,  71,  79,  87,  95, 103, 111, 119, 127,
   8, 17, 18, 35, 20, 37,  38,  71,  24,  41,  42,  75,  44,  77,  78, 143,
   9, 19, 21, 39, 25, 43,  45,  79,  41,  51,  53,  87,  57,  91,  93, 159,
  10, 21, 22, 43, 26, 45,  46,  87,  42,  53,  54,  91,  58,  93,  94, 175,
  11, 23, 27, 47, 43, 55,  59,  95,  75,  87,  91, 111, 107, 119, 123, 191,
  12, 25, 26, 51, 28, 53,  54, 103,  44,  57,  58, 107,  60, 109, 110, 207,
  13, 27, 29, 55, 45, 59,  61, 111,  77,  91,  93, 119, 109, 123, 125, 223,
  14, 29, 30, 59, 46, 61,  62, 119,  78,  93,  94, 123, 110, 125, 126, 239,
  15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255,
  16, 33, 34, 67, 36, 69,  70, 135,  40,  73,  74, 139,  76, 141, 142, 271,
...
From _Peter Munn_, Feb 24 2021: (Start)
We consider the case of n = 10, k = 41, following the procedure in the Feb 14 2021 comment.
First, write 10 and 41 in binary:
  10 = 1010_2
  41 = 101001_2
Add at least one leading zero to each number, equalizing number of zeros:
  0  0  1  0  1  0
  0  1  0  1  0  0  1
Align zeros, but separate ones:
  0     0  1     0  1  0
  |     |        |     |
  0  1  0     1  0     0  1
---------------------------
  0  1  0  1  1  0  1  0  1
Concatenating the ones, as shown above, we get 10110101_2 = 181.
So A(10, 41) = 181.
(End)

Cf. A005940, A156552.
Cf. A088698 (main diagonal).
Rows/columns 0-3: A001477, A005408, A341522, A004767. Row/column 7: A004771.
Cf. A341521 (the lower triangular section).
Cf. also A003991, A268725, A297164, A331590, A341509, A341510, A351960, A351961, A351962

Block[{nn = 12, a = {1}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Map[Prime[PrimePi[#1] + 1]^#2 & @@ # &, FactorInteger[#]] &@ a[[(i/2) + 1]], 2 a[[((i - 1)/2) + 1]]]], {i, nn}]; Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &[a[[1 + n - k]]*a[[1 + k]] ], {n, 0, nn}, {k, n, 0, -1}]] // Flatten (* Michael De Vlieger, Feb 24 2021 *)

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341520sq(n,k) = A156552(A005940(1+n)*A005940(1+k));
A341520list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A341520sq(col,(a-(col))))); (v); };
v341520 = A341520list(up_to);
A341520(n) = v341520[1+n];

A(x, y) = A156552(A005940(1+x) * A005940(1+y)).
For all n>=0, A(0, n) = A(n, 0) = n.
For all x>=0, y>=0, A(x, y) = A(y, x).
For all x, y, z >= 0, A(x, A(y, z)) = A(A(x, y), z).
From Antti Karttunen, Feb 27 2022: (Start)
For all x, y >= 0, A(x, y) = A(A351961(x,y), A351962(x,y)).
For x >= 0, y > 0, A(x, y) = A351960(x, A(x, A297164(y))).
(End)

A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13

Offset: 0

Antti Karttunen, Feb 12 2016

Each row n is row A006068(n) of array A268820 without its A006068(n) initial terms.

			The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
   1,  3,  6,  2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14
   2,  6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11
   3,  2,  7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10
   4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8, 24, 27, 25
   5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8, 25, 24
   6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
   7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24,  8
   8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
   9,  8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
  10, 11,  8,  9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
  11,  9, 24,  8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
  12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
  13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31, 29, 25, 27, 30, 26
  14, 10,  9, 11, 27, 25,  8, 24, 23, 21, 28, 20, 26, 30, 29, 31
  15, 14, 11, 10, 25, 24,  9,  8, 21, 20, 29, 28, 27, 26, 31, 30

Antti Karttunen, Table of n, a(n) for n = 0..15050; the first 173 antidiagonals of the array

Cf. A003188, A006068, A268714, A268820.
Main diagonal: A001969.
Row 0, column 0: A001477.
Row 1, column 1: A268717.
Antidiagonal sums: A268837.
Cf. A268719 (the lower triangular section).
Cf. also A268725.

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268715 n) (A268715bi (A002262 n) (A025581 n)))
(define (A268715bi row col) (A003188 (+ (A006068 row) (A006068 col))))
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi (A006068 row) (+ (A006068 row) col)))

A(i,j) = A003188(A006068(i) + A006068(j)) = A003188(A268714(i,j)).

A(row,col) = A268820(A006068(row), (A006068(row)+col)).

A268724 Square array A(i,j) = A006068(i) * A006068(j), read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 3, 2, 9, 2, 7, 6, 6, 7, 6, 21, 4, 21, 6, 4, 18, 14, 14, 18, 4, 5, 12, 12, 49, 12, 12, 5, 15, 15, 8, 42, 42, 8, 15, 15, 14, 45, 10, 28, 36, 28, 10, 45, 14, 12, 42, 30, 35, 24, 24, 35, 30, 42, 12, 13, 36, 28, 105, 30, 16, 30, 105, 28, 36, 13, 8, 39, 24, 98, 90, 20, 20, 90, 98, 24, 39, 8, 9, 24, 26, 84, 84, 60, 25, 60, 84, 84, 26, 24, 9

Offset: 1

Antti Karttunen, Feb 13 2016

			The top left [1 .. 15] x [1 .. 15] section of the array:
   1,  3,  2,  7,   6,  4,  5,  15,  14,  12,  13,   8,   9,  11,  10
   3,  9,  6,  21, 18, 12, 15,  45,  42,  36,  39,  24,  27,  33,  30
   2,  6,  4,  14, 12,  8, 10,  30,  28,  24,  26,  16,  18,  22,  20
   7, 21, 14,  49, 42, 28, 35, 105,  98,  84,  91,  56,  63,  77,  70
   6, 18, 12,  42, 36, 24, 30,  90,  84,  72,  78,  48,  54,  66,  60
   4, 12,  8,  28, 24, 16, 20,  60,  56,  48,  52,  32,  36,  44,  40
   5, 15, 10,  35, 30, 20, 25,  75,  70,  60,  65,  40,  45,  55,  50
  15, 45, 30, 105, 90, 60, 75, 225, 210, 180, 195, 120, 135, 165, 150
  14, 42, 28,  98, 84, 56, 70, 210, 196, 168, 182, 112, 126, 154, 140
  12, 36, 24,  84, 72, 48, 60, 180, 168, 144, 156,  96, 108, 132, 120
  13, 39, 26,  91, 78, 52, 65, 195, 182, 156, 169, 104, 117, 143, 130
   8, 24, 16,  56, 48, 32, 40, 120, 112,  96, 104,  64,  72,  88,  80
   9, 27, 18,  63, 54, 36, 45, 135, 126, 108, 117,  72,  81,  99,  90
  11, 33, 22,  77, 66, 44, 55, 165, 154, 132, 143,  88,  99, 121, 110
  10, 30, 20,  70, 60, 40, 50, 150, 140, 120, 130,  80,  90, 110, 100

Antti Karttunen, Table of n, a(n) for n = 1..15051; the first 173 antidiagonals of the array

Cf. A268725.
Cf. A006068 (row 1, column 1).
Cf. A268716 (row 3, column 3).
Cf. A268721 (the antidiagonal sums).
Cf. also A268714.

(define (A268724 n) (A268724bi (A002260 n) (A004736 n)))
(define (A268724bi row col) (* (A006068 row) (A006068 col)))

A(i,j) = A006068(i) * A006068(j)

A(i,j) = A006068(A268725(i,j)).

A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 2, 13, 5, 31, 27, 10, 8, 59, 63, 54, 52, 20, 22, 49, 17, 115, 119, 126, 124, 108, 110, 121, 105, 40, 42, 37, 45, 103, 99, 34, 32, 227, 231, 238, 236, 252, 254, 233, 249, 216, 218, 213, 221, 247, 243, 210, 208, 80, 82, 93, 85, 79, 75, 90, 88, 203, 207, 198, 196, 68, 70, 193, 65

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A003188, A006068.

Row 2 and column 2 of array A268725.

def A268722(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=3*k)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268722 n) (A003188 (* 3 (A006068 n))))

a(n) = A003188(3*A006068(n)).

cp's OEIS Frontend

A268723 Main diagonal of A268725: a(n) = A003188(A006068(n)^2), where A003188 is binary Gray code and A006068 is its inverse.

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A341520 Square array A(n,k) = A156552(A005940(1+n)*A005940(1+k)), read by antidiagonals.

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A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A268724 Square array A(i,j) = A006068(i) * A006068(j), read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

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