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

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

0,4

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of the array

Index entries for sequences related to binary expansion of n

Formula

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)

Example

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)

Mathematica

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 *)

Prog

(PARI)
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];

Crossrefs

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

Sequence in context: A236483 A266714 A151570 * A059036 A184442 A295609

Adjacent sequences: A341517 A341518 A341519 * A341521 A341522 A341523

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 13 2021

Status

approved