Can you name off the cubes whose base number exponents have their digits sum up to the cube of the base number? (yes undefined, this is in OEIS)

Moderator edit:Approved challenge question.

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- Nov 20th 2010, 06:26 AMwonderboy1953Cubes upon cubes
Can you name off the cubes whose base number exponents have their digits sum up to the cube of the base number? (yes undefined, this is in OEIS)

**Moderator edit:**Approved challenge question. - Nov 20th 2010, 07:10 AMtonio

It must be my three working neurons took some days (quite a few, in fact) off: I don't understand the given data:

"Cubes" (I suppose integer numbers which are cubes of other integers, like 8, 27, 64 and etc.), "whose base numbers

exponents"...what exponents? Of whom or what? For example, with $\displaystyle 64=4^3$ , are we talking of 3 (the exponent of 4)? That seems

to be a triviality, since all the numbers are cubes.

Tonio - Nov 20th 2010, 07:18 AMwonderboy1953
- Nov 20th 2010, 11:27 AMtonio
- Nov 20th 2010, 11:53 AMBruno J.
I agree with tonio, this question is very badly worded. Perhaps you should give an example of whatever it is you mean.

- Nov 20th 2010, 08:30 PMSoroban
Hello, wonderboy1953!

Since it is so*obvious*, maybe Mr. Fantastic can explain it to us.

Quote:

Can you name off the cubes whose base number exponents

have their digits sum up to the cube of the base number?

Here's my interpretation . . .

We have a cube, say, $\displaystyle 3^3 = 27.$

Write 27 in base-two: .$\displaystyle 27 \;=\;11011_2$

. . That is: .$\displaystyle 27 \;=\;2^4 + 2^3 + 2^1 + 2^0 $

The "base number exponents" are: .$\displaystyle 4, 3, 1, 0$

And these digits sum up to 8, the cube of the base number, 2.

We have a cube, say, $\displaystyle 16^3 = 4096$

Write 4096 in base-three: .$\displaystyle 4096 \:=\:12,\!121,\!201_3$

. . That is: .$\displaystyle 4096 \;=\;1\!\cdot\!3^7 + 2\!\cdot\!3^6 + 1\!\cdot\!3^5 + 2\!\cdot\!3^4 + 1\!\cdot\!3^3 + 2\!\cdot\!3^2 + 1\!\cdot\!3^0 $

The "base number exponents" are: .$\displaystyle 7, 6, 5, 4, 3, 2, 0$

And these digits sum up to 27, the cube of the base number, 3.

So far, I've found two of them . . . right?

- Nov 21st 2010, 08:42 AMwonderboy1953More information
Good try Soroban

All of the answers are in base 10 (and there are six answers).

I give Mr. Fantastic permission to restate the problem in better wording to clarify.

Somebody with a computer should be able to work it out in an hour.

Good luck to everybody.

Wonderboy1953 - Nov 21st 2010, 02:49 PMBruno J.
- Nov 21st 2010, 03:15 PMAlso sprach Zarathustra
- Nov 21st 2010, 03:28 PMtonio
- Nov 21st 2010, 03:30 PMAlso sprach Zarathustra
- Nov 22nd 2010, 07:44 AMwonderboy1953
Here's what I mean. I didn't specify what base the answers are to be expressed in (even though normally base 10 is assumed).

So technically Soroban is within his rights and he gave it a good try to meet the challenge.

I figured out a way to make my challenge clear by using a counterexample. Let's say we take the cube of 43 which is

79507. When you add the digits you get 28. Is 28 a cube? No.

Now my challenge is to take a number, cube it, and then add up its digits to see if you get another cube number from the digit summands. What numbers are these (my understanding is there are only six candidates which are in the OEIS). So get your computers ready to find these numbers.

(btw a secondary challenge would be to prove there are only six numbers that fill the bill and my last question is what are these numbers called).

Again good luck everyone.

PS The number you get after adding up the summands can't be just any cube, but the number you started off with when you were cubing. So if it were true that you got 43 instead of 28, then you have one of the answers which didn't turn out to be the case. - Nov 22nd 2010, 08:44 AMtonio

Finally, an explanation. So the challenge is for computers and/or programmers: what are the numbers such that when cubed the sum of the cube's digits equals a cube?

For example, 0, 1, 2, 5, 8, 10, 100, 1000, 10000, 100000, .....I see infinite numbers like these.

What had exponents and "sum up to the cube of the base number" to do with the above could be another challenge in the "semantics and weird wording" section, imho.

Tonio - Nov 24th 2010, 03:41 AMtonio
- Nov 24th 2010, 07:55 AMwonderboy1953
This part is correct: "Finally, an explanation. So the challenge is for computers and/or programmers: what are the numbers such that when cubed the sum of the cube's digits equals a cube?" but note my PS: "PS The number you get after adding up the summands can't be just any cube, but the number you started off with when you were cubing. So if it were true that you got 43 instead of 28, then you have one of the answers which didn't turn out to be the case." So that cube must equal the cube you started off with (and there are six such cubes).

How much progress have you made?