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iterated totient function∗
CompositeFan†
2013-03-21 21:46:33
The iterated totient function φk (n) is ak in the recurrence relation a0 = n
and ai = φ(ai−1 ) for i > 0, where φ(x) is Euler’s totient function.
After enough iterations, the function eventually hits 2 followed by an infinite
trail of ones. Ianucci et al define the “class” c of n as the integer such that
φc (n) = 2.
When the iterated totient function is summed thus:
c+1
X
φi (n)
i=1
x
it can be observed that just as 2 is a quasiperfect number when it comes to
adding up proper divisors, it is also “quasiperfect” when adding up the iterated
totient function. Quite unlike regular perfect numbers, 3x (which are obviously
odd) are “perfect” when adding up the iterated totient.
References
[1] D. E. Ianucci, D. Moujie & G. L. Cohen, “On Perfect Totient Numbers”
Journal of Integer Sequences, 6, 2003: 03.4.5
[2] R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag
2004: B42
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