A376243 -id:A376243 - cp's OEIS frontend

A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = xyz, where f(m) = A002487(m)/A002487(m+1).

0, 1, 2, 1, 6, 2, 2, 14, 4, 30, 12, 35, 2, 4, 9, 20, 4, 8, 126, 56, 32, 152, 52, 254, 61, 84, 40, 16, 4, 510, 368, 320, 212, 48, 396, 72, 583, 1022, 792, 368, 98, 188, 340, 80, 583, 339, 140, 32, 233, 2046, 480, 384, 583, 2062, 852, 188, 328

Offset: 1

M. F. Hasler, Sep 16 2024

nonn

A376241 uses the Stern-Brocot sequence s = A002487 to enumerate all (nonnegative) rational x = s(n)/s(n+1) and similarly y = s(m)/s(m+1), WLOG m <= n, which yield an integer x*y*z = x+y+z with (necessarily) z = (x+y)/(xy-1). The present sequence lists the m-values corresponding to the n-values listed in A376241.

			The terms correspond to the following solutions, with y = A002487(m)/A002487(m+1):
   m |  x  |  y  |  z  | xyz = x+y+z
-----+-----+-----+-----+------------
   0 |  0  |  0  |  0  |   0
   1 |  2  |  1  |  3  |   6
   2 | 3/2 | 1/2 | -8  |  -6
   1 |  3  |  1  |  2  |   6
   6 | 4/3 | 2/3 | -18 |  -16
   2 | 5/2 | 1/2 |  12 |   15
   2 |  4  | 1/2 | 9/2 |   9
  14 | 5/4 | 3/4 | -32 |  -30
  ...| ... | ... | ... |  ...

Cf. A002487 (Stern-Brocot sequence), A376241 (corresponding n values), A376243 (set of absolute values of corresponding xyz = x+y+z).

A376242(n, k=A376241(n))={my(p, q=1, x=A002487(k)/A002487(k+1)); for(m=2, k, my(y=(p=q)/q=A002487(m)); x*y != 1 && denominator(x+y+(x+y)/(x*y-1))==1 && return(m-1))} \\ Short of a function A376241(n), one can simply provide a term k = A376241(n) as second argument and omit the first argument n.

A376241 Indices k such that there exists m <= k such that x+y+z = xyz is an integer for x = f(k) := A002487(k)/A002487(k+1), y = f(m) and z = (x+y)/(xy-1).

Original entry on oeis.org

0, 3, 5, 7, 9, 11, 15, 17, 27, 33, 43, 44, 47, 55, 65, 107, 111, 119, 129, 135, 159, 167, 171, 257, 258, 427, 439, 495, 511, 513, 527, 575, 683, 751, 947, 951, 961, 1025, 1127, 1167, 1181, 1539, 1707, 1775, 1797, 1836, 1971, 2015, 2022, 2049, 2079, 2175, 2232, 2289, 2731, 3395, 3511

Offset: 1

M. F. Hasler, Sep 16 2024

This uses the Stern-Brocot sequence s = A002487 to enumerate all (nonnegative) rational x = s(n)/s(n+1) and similarly y = s(m)/s(m+1) (WLOG m <= n) which yield a rational solution {x, y, z} for the "Sum equals product problem", x*y*z = x+y+z = integer. The equality implies that z = (x+y)/(xy-1).

(z may be negative for negative integer solutions, which correspond to positive solutions if all the signs of (x, y, z) are flipped.

			The terms correspond to the following solutions, with x = A002487(k)/A002487(k+1):
   k |  x  |  y  |  z  | xyz = x+y+z
  ---+-----+-----+-----+------------
   0 |  0  |  0  |  0  |   0
   3 |  2  |  1  |  3  |   6
   5 | 3/2 | 1/2 | -8  |  -6
   7 |  3  |  1  |  2  |   6
   9 | 4/3 | 2/3 | -18 |  -16
  11 | 5/2 | 1/2 |  12 |   15
  15 |  4  | 1/2 | 9/2 |   9
  17 | 5/4 | 3/4 | -32 |  -30

Cf. A002487 (Stern-Brocot sequence), A376242 (corresponding m values), A376243 (set of absolute values of corresponding xyz = x+y+z).

is_A376241(n)={my(p, q=1, x=A002487(n)/A002487(n+1)); !n|| for(m=2, n, my(y=(p=q)/q=A002487(m)); x*y != 1 && denominator(x+y+(x+y)/(x*y-1))==1 && return(y))} \\ Return y=f(m) with the least possible m>0 such that x=f(n) and z=(x+y)/(xy-1) yield integer xyz = x+y+z, else zero.

cp's OEIS Frontend

A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = xyz, where f(m) = A002487(m)/A002487(m+1).

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A376241 Indices k such that there exists m <= k such that x+y+z = xyz is an integer for x = f(k) := A002487(k)/A002487(k+1), y = f(m) and z = (x+y)/(xy-1).

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cp's OEIS Frontend

A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = x*y*z, where f(m) = A002487(m)/A002487(m+1).

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PARI

A376241 Indices k such that there exists m <= k such that x+y+z = x*y*z is an integer for x = f(k) := A002487(k)/A002487(k+1), y = f(m) and z = (x+y)/(xy-1).

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Programs

PARI

A376242 a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = xyz, where f(m) = A002487(m)/A002487(m+1).

A376241 Indices k such that there exists m <= k such that x+y+z = xyz is an integer for x = f(k) := A002487(k)/A002487(k+1), y = f(m) and z = (x+y)/(xy-1).