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

Author

M. F. Hasler, Sep 16 2024

Keywords

• nonn
• more

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.

Examples

			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
		

Crossrefs

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

Programs

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.