The Lemon Fool

Find all numbers with the property that each is equal to the sum of the cubes of its digits. There aren’t many.

Cinelli

cinelli wrote:Find all numbers with the property that each is equal to the sum of the cubes of its digits. There aren’t many.

Cinelli

There are many. Indeed infinitely many!

Single digits: 1, 0, -1.

From these, 00, 01, 000, 001, 0000, 0001, etc.

But I expect you were looking at cases like
153 = 1 + 125 + 27
370 = 9 + 343 + 0
371 = 9 + 343 + 1
407 = 64 + 0 + 343

I found six, and this is all there are.

I have not counted -1 as its digit is 1. If you count negatives, there are five more, e.g., -371 is -3 hundreds, -7 tens and -1 unit.

407 is the same number as 00407.


Julian F. G. W.

jfgw wrote:I have not counted -1 as its digit is 1. If you count negatives, there are five more, e.g., -371 is -3 hundreds, -7 tens and -1 unit.

Agreed, playing with negatives (or complex numbers) here is a bit dodgy

407 is the same number as 00407.

But the question said nothing about distinct numbers!

UncleEbenezer wrote:But the question said nothing about distinct numbers!

I will let the OP ajudicate.

I would question infinity (expressed as an infinite sequence of digits) too.


Julian F. G. W.

Well solved, UncleEbenezer. A timely solution. This isn't a criticism but I believe several would be interested in the method you used.

Cinelli