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).

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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..57.

Example

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

Prog

(PARI) 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.

Crossrefs

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

Sequence in context: A331588 A077799 A260124 * A201646 A201647 A201648

Adjacent sequences: A376238 A376239 A376240 * A376242 A376243 A376244

Keyword

nonn,more

Author

M. F. Hasler, Sep 16 2024

Status

approved