A351786 -id:A351786 - cp's OEIS frontend

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

0, 1, 1, 3, 2, 3, 5, 7, 1, 5, 3, 7, 9, 13, 11, 15, 4, 5, 9, 11, 6, 7, 13, 15, 17, 21, 19, 23, 25, 29, 27, 31, 1, 9, 5, 13, 17, 25, 21, 29, 3, 11, 7, 15, 19, 27, 23, 31, 33, 41, 37, 45, 49, 57, 53, 61, 35, 43, 39, 47, 51, 59, 55, 63, 8, 9, 17, 19, 10, 11, 21

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) = 13.

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. A000120, A000695, A072345, A351706, A351785, A351786.

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

a(A000695(n)) = n.
a(2^k) = A072345(k-1) for any k > 0.
a(2^k-1) = 2^k-1 for any k >= 0.
A000120(a(n)) = A000120(n).

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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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.

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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)).

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cp's OEIS Frontend

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

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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.

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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)).

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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)).