# Sequence A082897: Perfect Totient Numbers

The numbers in this sequence are like the perfect numbers, but adding together the iterates of the totient function instead of adding together the number's factors.

The Euler totient function, Sloane's A0010, counts how many numbers smaller than N are relatively prime to N. For example, if N is 14, the numbers 1, 3, 5, 9, 11 and 13 are relatively prime (notice that 1 counts), so the totient function of 14 is 6. For a prime P we're counting all numbers smaller than P, which is P-1, so the totient function always gives a smaller number except (by definition) when N=1.

So, we can take a number like 243, and compute the Totient function (which gives 162), then compute the totient function of that number, and so on (giving 54, 18, 6, 2 and 1), then add all these numbers together and we get 162 + 54 + 18 + 6 + 2 + 1 = 243. That makes it a "perfect totient number". The powers of 3 are all perfect totient numbers (fairly easy to show; see Wikipedia or Sloane's entry for A82897). These are labeled p3 below.

We also see that some numbers of the form 2^{2N}-1 (like 15, 255,
65535, and 4292967295) are perfect totient numbers. These are labeled
2n1 below. However, 2^{26}-1=18446744073709551615 is not a
perfect totient number.

If a prime P is of the form P=4×3^{N}+1 for some integer N,
then 3P is a perfect totient number[1]. The primes
P of this form are: 5, 13, 37, 109, 2917, 19131877, 57395629,
16210220612075905069, ... (Sloane's A125734, with
values of N from A5537), giving the perfect
totient numbers 3P: 15, 39, 111, 327, 8751,
57395631, 172186887, 48630661836227715207, ... (not currently in
OEIS). These are labeled 43n1 in the list below.

Here is a table of the perfect totient numbers as far as I've been
able to compute. Each line gives the index N, the number A_{N},
and its factorization. After some of the factorizations are markers
indicating if the number belongs to one of the two classes identified
above: p3 for powers of 3, 2n1 for the 2^{N}-1 type, and
43n1 for the 3×(4×3^{N}+1) type:

A_{1} = 3 = 3 p3 2n1

A_{2} = 9 = 3^{2} p3

A_{3} = 15 = 3×5 43n1 2n1

A_{4} = 27 = 3^{3} p3

A_{5} = 39 = 3×13 43n1

A_{6} = 81 = 3^{4} p3

A_{7} = 111 = 3×37 43n1

A_{8} = 183 = 3×61

A_{9} = 243 = 3^{5} p3

A_{10} = 255 = 3×5×17 2n1

A_{11} = 327 = 3×109 43n1

A_{12} = 363 = 3×11^{2}

A_{13} = 471 = 3×157

A_{14} = 729 = 3^{6} p3

A_{15} = 2187 = 3^{7} p3

A_{16} = 2199 = 3×733

A_{17} = 3063 = 3×1021

A_{18} = 4359 = 3×1453

A_{19} = 4375 = 5^{4}×7

A_{20} = 5571 = 3^{2}×619

A_{21} = 6561 = 3^{8} p3

A_{22} = 8751 = 3×2917 43n1

A_{23} = 15723 = 3^{2}×1747

A_{24} = 19683 = 3^{9} p3

A_{25} = 36759 = 3×12253

A_{26} = 46791 = 3^{3}×1733

A_{27} = 59049 = 3^{10} p3

A_{28} = 65535 = 3×5×17×257 2n1

A_{29} = 140103 = 3^{3}×5189

A_{30} = 177147 = 3^{11} p3

A_{31} = 208191 = 3×29×2393

A_{32} = 441027 = 3^{2}×49003

A_{33} = 531441 = 3^{12} p3

A_{34} = 1594323 = 3^{13} p3

A_{35} = 4190263 = 7×11×54419

A_{36} = 4782969 = 3^{14} p3

A_{37} = 9056583 = 3^{3}×335429

A_{38} = 14348907 = 3^{15} p3

A_{39} = 43046721 = 3^{16} p3

A_{40} = 57395631 = 3×19131877 43n1

A_{41} = 129140163 = 3^{17} p3

A_{42} = 172186887 = 3×57395629 43n1

A_{43} = 236923383 = 3×1427×55343

A_{44} = 387420489 = 3^{18} p3

A_{45} = 918330183 = 3^{3}×34012229

A_{46} = 1162261467 = 3^{19} p3

A_{47} = 3486784401 = 3^{20} p3

A_{48} = 3932935775 = 5^{2}×29×5424739

A_{49} = 4294967295 = 3×5×17×257×65537 2n1

A_{50} = 4764161215 = 5×11×86621113

A_{51} = 9158284383 = 3×101^{2}×299261

A_{52} = 10460353203 = 3^{21} p3

A_{53} = 10774273279 = 7×11×47×509×5849

A_{54} = 31381059609 = 3^{22} p3

[1] T. Venkataraman, "Perfect totient number". The Mathematics Student 43 178 (1975). MR0447089.

Some other sequences are discussed here.

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