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perfect totient number (Definition)

An integer $ n$ is a perfect totient number if

$\displaystyle n = \sum_{i = 1}^{c + 1} \phi^i(n)$
, where $ \phi^i(x)$ is the iterated totient function and $ c$ is the integer such that $ \phi^c(n) = 2$.

A082897 in Sloane's OEIS lists the first few perfect totient numbers: 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, etc. It can be observed that many of these are multiples of 3 (in fact, 4375 is the smallest one that is not divisible by 3) and in fact all $ 3^x$ for $ x > 0$ are perfect totient numbers.

Furthermore, $ 3p$ for a prime $ p > 3$ is a perfect totient number if and only if $ p = 4n + 1$, where $ n$ itself is also a perfect totient number. Mohan and Suryanarayana showed why $ 3p$ can't be a perfect totient number when $ p \equiv 3 \mod 4$. In regards to $ 3^2p$, Ianucci et al showed that if it is a perfect totient number then $ p$ is a prime of one of three specific forms listen in their paper. It is not known if there are any perfect totient numbers of the form $ 3^xp$ for $ x > 3$.

Bibliography

1
Perez Cacho, ``On the sum of totients of successive orders,'' Revista Matematica Hispano-Americana 5.3 (1939): 45 - 50
2
D. E. Ianucci, D. Moujie & G. L. Cohen, ``On Perfect Totient Numbers'' Journal of Integer Sequences, 6, 2003: 03.4.5
3
R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: B42



"perfect totient number" is owned by CompositeFan.
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See Also: iterated totient function

Other names:  totient perfect number

Attachments:
proof that 3 is the only prime perfect totient number (Proof) by PrimeFan
proof that all powers of 3 are perfect totient numbers (Proof) by PrimeFan
examples of perfect totient numbers (Example) by PrimeFan
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Cross-references: prime, divisible, multiples, OEIS, iterated totient function, integer
There are 5 references to this entry.

This is version 3 of perfect totient number, born on 2007-01-12, modified 2007-01-16.
Object id is 8741, canonical name is PerfectTotientNumber.
Accessed 1075 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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Question about 4n + 1 by PrimeFan on 2007-01-13 16:53:40
Apologies in advance for the tenor of this question. In all likelihood it will appear to be an extremely gauche question of little importance distracting from topics considered more important, such as the curvature of space-time with an irrational number of dimensions. To me right now, it seems that the answer is obvious but I can't figure it out for some reason.

If an integer is of the form 4n + 1, doesn't that automatically rule out the possibility that it could be equiv 3 mod 4, that is, 4n - 1?
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