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 Function Function Function Function Foray (Posted on 2015-03-31)
N is a positive integer and F(N) denotes the sum of the base ten digits of N.

Find F(F(F(22016))) and F(F(F(F(22016))))

*** As an extra challenge, solve the puzzle without using a computer program assisted method.

 No Solution Yet Submitted by K Sengupta No Rating

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 A good guess / computer help | Comment 1 of 2
The process of repeatedly applying F eventually gives the digital root of 2^2016.
Powers of 2 have digital roots in a cycle of length 6.  2016 is divisible by 6 so the digital root of 2^2016 = the digital root of 2^0 = 1.
The question is then whether F has been applied enough times.

F(2^2016) should have be about 4.5*2016*log(2) = 2731.
F(F(2^2016)) will be a two digit number [unless F(2^2016) happens to be a power of 10]
F(F(F(2^2016))) will be 10
F(F(F(F(2^2016)))) will be 1.

Checking
https://oeis.org/A001370/b001370.txt
gives
F(2^2016) = 2656
F(F(2^2016)) = 19
F(F(F(2^2016))) = 10
F(F(F(F(2^2016)))) = 1

 Posted by Jer on 2015-03-31 12:50:47

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