Three 3-digit non leading zero positive base N integers

**P**,

**Q**
and

**R**, with

**P** >

**Q** >

**R**, are such that:

(i)

**Q** is the harmonic mean of

**P** and

**R**, and:

(ii)

**P**,

**Q** and

**R** can be derived from one another by

**cyclic permutation ** of digits.

Determine all possible positive integer values of N ≤ 30 for which this is possible.

I can't come up with anything using the following code:

FOR n = 2 TO 30

FOR a = 1 TO n-1

FOR b = 0 TO n-1

FOR c = 0 TO n-1

abc = Base2Dec(a, b, c, n)

bca = Base2Dec(b, c, a, n)

cab = Base2Dec(c, a, b, n)

IF abc <> bca AND abc <> cab THEN

IF bca = HMean(abc, cab) THEN PRINT abc, bca, cab, n

IF cab = HMean(abc, bca) THEN PRINT abc, cab, bca, n

IF abc = HMean(bca, cab) THEN PRINT bca, abc, cab, n

END IF

NEXT c

NEXT b

NEXT a

NEXT n

END

FUNCTION Base2Dec (a, b, c, base)

Base2Dec = a*base^2 + b*base + c

END FUNCTION

FUNCTION HMean (x, y)

HMean = 2*x*y / (x + y)

END FUNCTION

The closest I can find is '102', '120', and '210' in base 4 (18, 24, and 36 respectively), which I'm pretty sure is not a **cyclic** permutation.

I also ran the above code for n = 31 to 50 and still came up with nothing.