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May be this problem is already answered. I am not aware.

What is the next number in the following series?

Importantly, after that number, are there any more numbers? is there a math related reason for that?

1, 512, 4913, 5832, 17576, ?

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I believe these are the numbers known as:

Dudeney numbers - in particular, the sequence of base-10 natural numbers (hence no 0) whose digits sum to the number's cube root.

Like so:

1 (trivially)
512 = 8 cubed and digit sum 5+1+2=8
4913 = 17 cubed and digit sum 4+9+1+3=17
5832 = 18 cubed and digit sum 5+8+3+2=18
17576 = 26 cubed and digit sum 1+7+5+7+6=26

The next number in the sequence is:

19683 which both is 27 cubed and has digit sum 27.

Moreover:

These have been demonstrated to be the only six numbers to possess this property in base-10. See the linked Wiki page for a discussion - I will leave any further exposition of the mathematics to another puzzler who has the stamina to engage in a MathJax marathon!

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  • $\begingroup$ I think that zero should be added as well? $\endgroup$ – LeppyR64 Jun 18 '20 at 16:43
  • $\begingroup$ There is a 7th number, 0 (which of course comes before the 1 and is usually left out because it's trivial) $\endgroup$ – Bennett Bernardoni Jun 18 '20 at 16:43
  • $\begingroup$ @BennettBernardoni ninja'd :) $\endgroup$ – LeppyR64 Jun 18 '20 at 16:43
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    $\begingroup$ Lol, by 4 seconds. So close $\endgroup$ – Bennett Bernardoni Jun 18 '20 at 16:44
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    $\begingroup$ @LeppyR64 (and Bennett!) That's true - I should specify we're talking natural numbers only. Thanks... $\endgroup$ – Stiv Jun 18 '20 at 16:49

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