Let $ n > 1 $ be an integer.
Consider The prime factorization
$$ n = x_1 x_2 x_3 ... $$
Now define
$$ t(n) = t( x_1 x_2 x_3 ...) = t(x_1) + t(x_2) + t(x_3) + ... + t( x_1 + x_2 + x_3 + ... ) $$
Clearly this function is completely determined by its values at primes. This gives us multiple solutions.
Im fascinated by this function.
Has this been studied before ?
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Some more specific questions :
Can we find solutions such that
$$ t(n) \leq t(n+1) \leq t(n+2) $$
And if so , which one grows the slowest ? And how fast does that slowest solution grow asymptotically ?
How about The fastest ?
Are they easy questions or do they depend on a lot of theory or open conjectures ?
A reference or a few plots would be nice too.
Im not sure how to handle this.
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Update :
There seems to be a problem with defining $t(2n)$ so for now i focus on odd $n$.
From $t(3^{3^k}) $ we conclude that f grows like $ O( ln(n) + ln(ln(n)) + ln(ln(ln(n))) + ... ) $