A355090 -id:A355090 - cp's OEIS frontend

A357144 Square array, A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) * f(k)) where f(m) = A002487(m)/A002487(m+1) and g is the inverse of f.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 1, 1, 4, 0, 0, 5, 32, 15, 32, 5, 0, 0, 6, 14, 6, 6, 14, 6, 0, 0, 7, 4, 7, 256, 7, 4, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 128, 63, 48, 35, 48, 63, 128, 9, 0, 0, 10, 6, 2, 1, 1, 1, 1, 2, 6, 10, 0, 0, 11, 56, 27, 2048, 47, 60, 47, 2048, 27, 56, 11, 0

Offset: 0

Rémy Sigrist, Sep 15 2022

The function f is a bijection from the nonnegative integers to the nonnegative rational numbers.
The positive integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the multiplicative group of positive rational numbers (f and g act as isomorphisms).
Each row (or column), except the first, is a permutation of the nonnegative integers.

			Array A(n, k) begins:
  n\k | 0   1    2   3     4     5    6    7      8    9    10   11    12
  ----+------------------------------------------------------------------
    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
    2 | 0   2    8   1    32    14    4    5    128    6    56   17    16
    3 | 0   3    1  15     6     7    9   63      2   27    33   31    30
    4 | 0   4   32   6   256     2   48    1   2048   60    16   62   384
    5 | 0   5   14   7     2    35    1   47     20    3  1022  119    10
    6 | 0   6    4   9    48     1   60    3     32  510    12   13    72
    7 | 0   7    5  63     1    47    3  511     14   15    61  383    33
    8 | 0   8  128   2  2048    20   32   14  32768    4   320   26   512
    9 | 0   9    6  27    60     3  510   15      4   93    30   39   258
   10 | 0  10   56  33    16  1022   12   61    320   30   196    5  1008
   11 | 0  11   17  31    62   119   13  383     26   39     5  575     1
   12 | 0  12   16  30   384    10   72   33    512  258  1008    1   960

Rémy Sigrist, PARI program

Cf. A002487, A054429, A354522, A355090.

PARI
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See Links section.
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A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, 1) = n.
A(n, A054429(n)) = 1 for any n > 0.
A(m, A(n, k)) = A(A(m, n), k).
A(n, A(n-1, ... A(2, 1) ... )) = 2^(A002487(n+1)-1).

A358110 Indices of the harmonic numbers in the Stern-Brocot sequence (A002487).

Original entry on oeis.org

0, 1, 5, 125, 8195, 32675, 755, 34763, 520283, 37773179, 21743337467, 4647489635464983347207, 1236947931143, 272658152711, 604398345569737906323527, 9595849053479089263878087, 3693713292455, 288389531265129191, 11150032316898390632304469945009811031588839

Offset: 0

Peter Luschny, Nov 08 2022

nonn

We assume the harmonic numbers to start with H(0) = 0.

			Let Fusc(n) = fusc(n) / fusc(n + 1) where fusc = A002487.
          0 = H(0) = Fusc(0)      => a(0) = 0.
          1 = H(1) = Fusc(1)      => a(1) = 1.
      (3/2) = H(2) = Fusc(5)      => a(2) = 5.
     (11/6) = H(3) = Fusc(125)    => a(3) = 125.
    (25/12) = H(4) = Fusc(8195)   => a(4) = 8195.
   (137/60) = H(5) = Fusc(32675)  => a(5) = 32676.
    (49/20) = H(6) = Fusc(755)    => a(6) = 755.
  (363/140) = H(7) = Fusc(34763)  => a(7) = 34763.
  (761/280) = H(8) = Fusc(520283) => a(8) = 520283.

Edsger W. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence A002487 is called fusc).

Edsger W. Dijkstra, More about the function "fusc".
Peter Luschny, Rational Trees and Binary Partitions.
Rémy Sigrist, Logarithmic binary plot of the sequence for n = 0..2048

Cf. A002487, A001008/A002805 (harmonic), A355090.

a(n) = { my (h=sum(i=1, n, 1/i), x=numerator(h), y=denominator(h)); if (x==0, 0, my (v=0, t=1, a=0, b=1, c=1, d=0); while (1, my (m=a+c, n=b+d); if (x*n==y*m, return (t+v), x*nRémy Sigrist, Nov 08 2022

# using function harmonic from A001008
def A358110(n: int) -> int:
    if n == 0: return 0
    x, y = harmonic(1, n + 1)
    a = d = v = 0
    b = c = t = 1
    while True:
        m = a + c
        n = b + d
        if x * n == y * m:
            return v + t
        if x * n < y * m:
            c, d = m, n
        else:
            v, a, b = v + t, m, n
        t *= 2
print([A358110(n) for n in range(19)]) # (after Rémy Sigrist) Peter Luschny, Nov 08 2022

a(n) = A355090(A001008(n), A002805(n)) for any n > 0. - Rémy Sigrist, Nov 08 2022

More terms from Rémy Sigrist, Nov 08 2022

cp's OEIS Frontend

A357144 Square array, A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) * f(k)) where f(m) = A002487(m)/A002487(m+1) and g is the inverse of f.

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