I find Base-sensitive properties of numbers really interesting at times, and this is an example. For example, let f(x) be a recursive sequence where f(0)=2, and $f(x+1)=\digitsum(f(x)^n)$. If n equals 2, This sequence goes 2,4,7,13,16,13... and loops between 13 and 16. Starting with a different value, like f(0)=17, the sequence goes 17,19,10,1,1,1... . okay, one more time, let f(0)=13, and let's let n equal 3 this time. The sequence now becomes 13,19,28,19... . My question is, is it possible for any starting number x and power n to cause an infinite sequence that tends to infinity and doesn't form a loop?
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It will always eventually decrease. That is, your sequence will decrease when the number of digits of $x$ is much larger than $n$, because the digit sum of a base $b$ number in $[b^d, b^{d+1})$ is at most $(b-1)(d+1)$. – Vepir Mar 31 '22 at 13:20
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Ok, that is very interesting! Thank you – Catman 321 Apr 25 '22 at 18:32