A351705 -id:A351705 - cp's OEIS frontend

A351706 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the denominator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351705 for the numerators.

1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 2, 2, 1, 1, 2

Offset: 0

Rémy Sigrist, Feb 16 2022

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

			For n = 13:
- 13 = 2^0 + 2^2 + 2^3,
- A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
- d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
- so a(13) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Wikipedia, Dyadic rational
Index entries for sequences related to binary expansion of n

Cf. A000695, A016116, A072345, A351705, A351785, A351786.

a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n,2); d+=2^((-1)^k*(k+1)\2)); denominator(d) }

a(A000695(n)) = 1.
a(2^k) = A072345(k) for any k >= 0.
a(2^k-1) = A016116(k) for any k >= 0.

A351785 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) + d(k)).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 1, 6, 4, 5, 5, 4, 4, 5, 5, 6, 16, 6, 5, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 5, 18, 16, 18, 5, 18, 8, 9, 9, 16, 16, 17, 17, 16, 16, 9, 9, 10, 12, 10, 17, 18, 20, 18, 17, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11

Offset: 0

Rémy Sigrist, Feb 19 2022

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   4   3   6   5  16   7  18   9  12  11  14  13  24  15  26
    2|   2   3   1   4   6   7   5  16  10  11   9  12  14  15  13  24
    3|   3   6   4   5   7  18  16  17  11  14  12  13  15  26  24  25
    4|   4   5   6   7  16  17  18  19  12  13  14  15  24  25  26  27
    5|   5  16   7  18  17  20  19  22  13  24  15  26  25  28  27  30
    6|   6   7   5  16  18  19  17  20  14  15  13  24  26  27  25  28
    7|   7  18  16  17  19  22  20  21  15  26  24  25  27  30  28  29
    8|   8   9  10  11  12  13  14  15   2   3   1   4   6   7   5  16
    9|   9  12  11  14  13  24  15  26   3   6   4   5   7  18  16  17
   10|  10  11   9  12  14  15  13  24   1   4   3   6   5  16   7  18

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k))
Wikipedia, Dyadic rational
Index entries for sequences related to binary expansion of n

Cf. A000695, A130472, A351705, A351706, A351786 (multiplication).

d(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^((-1)^k*(k+1)\2)); v }
t(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^if (k>=0, 2*k, -1-2*k)); v }
T(n,k) = t(d(n)+d(k))

T(A000695(n), A000695(k)) = A000695(n + k).
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = n.

A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).

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, 1, 12, 1, 5, 0, 0, 6, 3, 5, 5, 3, 6, 0, 0, 7, 9, 18, 16, 18, 9, 7, 0, 0, 8, 11, 15, 20, 20, 15, 11, 8, 0, 0, 9, 32, 25, 17, 65, 17, 25, 32, 9, 0, 0, 10, 34, 40, 21, 23, 23, 21, 40, 34, 10, 0

Offset: 0

Rémy Sigrist, Feb 19 2022

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
  ---+-------------------------------------------------------------------------
    0|  0   0   0   0   0   0    0    0    0    0    0    0    0    0    0    0
    1|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
    2|  0   2   8  10   1   3    9   11   32   34   40   42   33   35   41   43
    3|  0   3  10  12   5  18   15   25   40   43   33   38   45   58   48   51
    4|  0   4   1   5  16  20   17   21    2    6    3    7   18   22   19   23
    5|  0   5   3  18  20  65   23   70   10   15   12   25   30   75   72   77
    6|  0   6   9  15  17  23   28   74   34   37   43   56   51   96   62  105
    7|  0   7  11  25  21  70   74   88   42   56   38   52   63  109   99  113
    8|  0   8  32  40   2  10   34   42  128  136  160  168  130  138  162  170
    9|  0   9  34  43   6  15   37   56  136  131  170  164  142  144  173  178
   10|  0  10  40  33   3  12   43   38  160  170  130  137  163  172  132  142

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k))
Wikipedia, Dyadic rational
Index entries for sequences related to binary expansion of n

Cf. A000695, A130472, A351705, A351706, A351785 (addition).

d(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^((-1)^k*(k+1)\2)); v }
t(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^if (k>=0, 2*k, -1-2*k)); v }
T(n,k) = t(d(n)*d(k))

T(A000695(n), A000695(k)) = A000695(n * k).
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = 0.
T(n, 1) = n.

cp's OEIS Frontend

A351706 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the denominator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351705 for the numerators.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351785 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) + d(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A351706 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the denominator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351705 for the numerators.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351785 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i*2^i, let d(m) = Sum_{i >= 0} b_i * 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) + d(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i*2^i, let d(m) = Sum_{i >= 0} b_i * 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351785 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) + d(k)).

A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i2^i, let d(m) = Sum_{i >= 0} b_i 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).