Consider decimal number 33.
If converted to base 4 form it would be 201, which if regarded as 7 base number would yield 99 decimal.
Pay attention: 99=3*33.
219 is another number displaying the same feature (when converted to base 4 and interpreted as base 7 creates a multiple of the original decimal number).
Please list some additional samples.
Any string of n digits written in base 7 instead of base 4 will be around (7/4)^(n1) times bigger.
I wrote a little program dividing the string of digits in base 7 and dividing by base 4 an noting when this ratio was an integer.
Base 10 base 4 / 7 base 10 ratio
33 > 201 > 99 ratio = 3
(7/4)^2 = 3.0625
219 > 3123 > 1095 ratio =5
(7/4)^3 = 5.359
840 > 31020 > 7560 ratio = 9
(7/4)^4 = 9.379
1055 > 100133 > 16880 ratio = 16
(7/4)^5 = 16.4
Searching up to 6 digits base 4
interestingly there is one solution for each length (none for length 2 or 1)

Posted by Jer
on 20151103 14:24:44 