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.
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.
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!