A351960 -id:A351960 - cp's OEIS frontend

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

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)

A297164 Permutation of nonnegative integers: a(n) = A156552(A005940(1+n)-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 5, 6, 17, 16, 23, 64, 65, 10, 9, 32, 19, 128, 129, 512, 67, 256, 47, 24, 4097, 20, 4099, 32768, 79, 1024, 11, 18, 31, 14, 29, 4096, 2049, 66, 515, 514, 263, 65536, 2053, 8388608, 524289, 16384, 87, 16777216, 2097153, 70, 524291, 17179869184, 287, 524288, 1037, 8388610, 289, 134217728, 1071, 520, 97, 38, 15, 12, 257, 8192, 63

Offset: 1

Antti Karttunen, Jan 05 2018

nonn

Note the indexing: although the domain starts from 1, the range includes also zero.

Inverse: A297163.
Cf. A005940, A156552.
Cf. also A297166, A341520, A351960.

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}; \\ From A156552
A297164(n) = A156552(A005940(1+n)-1); \\ Antti Karttunen, Feb 27 2022

(define (A297164 n) (A156552 (+ -1 (A005940 (+ 1 n)))))

a(n) = A156552(A005940(1+n)-1).

A351961 Square array A(n,k) = A156552(gcd(A005940(1+n), A005940(1+k))), read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 26 2022

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.

			The top left corner of the array:
   n=  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17
-----|--------------------------------------------------------------
k= 0 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,
   1 | 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,  0,  1,  0,  1,  0,  1,  0,  1,
   2 | 0, 0, 2, 0, 0, 2, 2, 0, 0, 0,  2,  2,  0,  2,  2,  0,  0,  0,
   3 | 0, 1, 0, 3, 0, 1, 0, 3, 0, 1,  0,  3,  0,  1,  0,  3,  0,  1,
   4 | 0, 0, 0, 0, 4, 0, 0, 0, 0, 4,  4,  0,  4,  0,  0,  0,  0,  0,
   5 | 0, 1, 2, 1, 0, 5, 2, 1, 0, 1,  2,  5,  0,  5,  2,  1,  0,  1,
   6 | 0, 0, 2, 0, 0, 2, 6, 0, 0, 0,  2,  2,  0,  6,  6,  0,  0,  0,
   7 | 0, 1, 0, 3, 0, 1, 0, 7, 0, 1,  0,  3,  0,  1,  0,  7,  0,  1,
   8 | 0, 0, 0, 0, 0, 0, 0, 0, 8, 0,  0,  0,  0,  0,  0,  0,  0,  8,
   9 | 0, 1, 0, 1, 4, 1, 0, 1, 0, 9,  4,  1,  4,  1,  0,  1,  0,  1,
  10 | 0, 0, 2, 0, 4, 2, 2, 0, 0, 4, 10,  2,  4,  2,  2,  0,  0,  0,
  11 | 0, 1, 2, 3, 0, 5, 2, 3, 0, 1,  2, 11,  0,  5,  2,  3,  0,  1,
  12 | 0, 0, 0, 0, 4, 0, 0, 0, 0, 4,  4,  0, 12,  0,  0,  0,  0,  0,
  13 | 0, 1, 2, 1, 0, 5, 6, 1, 0, 1,  2,  5,  0, 13,  6,  1,  0,  1,
  14 | 0, 0, 2, 0, 0, 2, 6, 0, 0, 0,  2,  2,  0,  6, 14,  0,  0,  0,
  15 | 0, 1, 0, 3, 0, 1, 0, 7, 0, 1,  0,  3,  0,  1,  0, 15,  0,  1,
  16 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16,  0,
  17 | 0, 1, 0, 1, 0, 1, 0, 1, 8, 1,  0,  1,  0,  1,  0,  1,  0, 17,

Index entries for sequences related to binary expansion of n

Cf. A003989, A005940, A156552.
Cf. A001477 (main diagonal).
Cf. also A341520, A351960, A351962.

up_to = 104; \\ 10439 = binomial(144+1,2)-1
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 };
A351961sq(n,k) = A156552(gcd(A005940(1+n),A005940(1+k)));
A351961list(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] = A351961sq(col,(a-(col))))); (v); };
v351961 = A351961list(up_to);
A351961(n) = v351961[1+n];

For all x, y >= 0, A(x, y) = A(x, A351960(x,y)) = A(A351960(x,y), y).

A351962 Square array A(n,k) = A156552(lcm(A005940(1+n), A005940(1+k))), read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 26 2022

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.

			The top left corner of the array:
      n=0   1   2   3   4   5    6    7    8   9   10   11   12
-----|-----------------------------------------------------------
k= 0 |  0,  1,  2,  3,  4,  5,   6,   7,   8,  9,  10,  11,  12,
   1 |  1,  1,  5,  3,  9,  5,  13,   7,  17,  9,  21,  11,  25,
   2 |  2,  5,  2, 11, 10,  5,   6,  23,  18, 21,  10,  11,  26,
   3 |  3,  3, 11,  3, 19, 11,  27,   7,  35, 19,  43,  11,  51,
   4 |  4,  9, 10, 19,  4, 21,  22,  39,  20,  9,  10,  43,  12,
   5 |  5,  5,  5, 11, 21,  5,  13,  23,  37, 21,  21,  11,  53,
   6 |  6, 13,  6, 27, 22, 13,   6,  55,  38, 45,  22,  27,  54,
   7 |  7,  7, 23,  7, 39, 23,  55,   7,  71, 39,  87,  23, 103,
   8 |  8, 17, 18, 35, 20, 37,  38,  71,   8, 41,  42,  75,  44,
   9 |  9,  9, 21, 19,  9, 21,  45,  39,  41,  9,  21,  43,  25,
  10 | 10, 21, 10, 43, 10, 21,  22,  87,  42, 21,  10,  43,  26,
  11 | 11, 11, 11, 11, 43, 11,  27,  23,  75, 43,  43,  11, 107,
  12 | 12, 25, 26, 51, 12, 53,  54, 103,  44, 25,  26, 107,  12,
  13 | 13, 13, 13, 27, 45, 13,  13,  55,  77, 45,  45,  27, 109,
  14 | 14, 29, 14, 59, 46, 29,  14, 119,  78, 93,  46,  59, 110,
  15 | 15, 15, 47, 15, 79, 47, 111,  15, 143, 79, 175,  47, 207,
  16 | 16, 33, 34, 67, 36, 69,  70, 135,  40, 73,  74, 139,  76,

Cf. A003990, A005940, A156552.
Cf. A001477 (main diagonal).
Cf. also A341520, A351960, A351961.

up_to = 104;
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 };
A351962sq(n,k) = A156552(lcm(A005940(1+n),A005940(1+k)));
A351962list(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] = A351962sq(col,(a-(col))))); (v); };
v351962 = A351962list(up_to);
A351962(n) = v351962[1+n];

cp's OEIS Frontend

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

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A297164 Permutation of nonnegative integers: a(n) = A156552(A005940(1+n)-1).

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A351961 Square array A(n,k) = A156552(gcd(A005940(1+n), A005940(1+k))), read by antidiagonals.

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A351962 Square array A(n,k) = A156552(lcm(A005940(1+n), A005940(1+k))), read by antidiagonals.

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