Perfect totient number

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number "n", apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals "n", then "n" is a perfect totient number. Or to put it algebraically, if:$n\; =\; sum\_\{i\; =\; 1\}^\{c\; +\; 1\}\; varphi^i(n),$where:is the iterated totient function and "c" is the integer such that:$displaystylevarphi^c(n)=2,$then "n" is a perfect totient number.

The first few perfect totient numbers are:3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... OEIS|id=A082897.

For example, start with 327. φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.

Multiples and powers of three

It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that:$displaystylevarphi(3^k)\; =\; varphi(2\; imes\; 3^k)\; =\; 2\; imes\; 3^\{k-1\}.$

Venkataraman (1975) found another family of perfect totient numbers: if "p" = 4×3^k+1 is prime, then 3"p" is a perfect totient number. The values of "k" leading to perfect totient numbers in this way are:0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... OEIS|id=A005537.

More generally if "p" is a prime number greater than three, and 3"p" is a perfect totient number, then "p" ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all "p" of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Ianucci et al. (2003) showed that if 9"p" is a perfect totient number then "p" is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers that are multiples of powers of 3 greater than 9 but not themselves powers of three.

References

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