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# 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 22N-1 (like 15, 255, 65535, and 4292967295) are perfect totient numbers. These are labeled 2n1 below. However, 226-1=18446744073709551615 is not a perfect totient number.

If a prime P is of the form P=4×3N+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 AN, 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 2N-1 type, and 43n1 for the 3×(4×3N+1) type:

A1 = 3 = 3    p3    2n1
A2 = 9 = 32    p3
A3 = 15 = 3×5    43n1    2n1
A4 = 27 = 33    p3
A5 = 39 = 3×13    43n1
A6 = 81 = 34    p3
A7 = 111 = 3×37    43n1
A8 = 183 = 3×61
A9 = 243 = 35    p3
A10 = 255 = 3×5×17    2n1
A11 = 327 = 3×109    43n1
A12 = 363 = 3×112
A13 = 471 = 3×157
A14 = 729 = 36    p3
A15 = 2187 = 37    p3
A16 = 2199 = 3×733
A17 = 3063 = 3×1021
A18 = 4359 = 3×1453
A19 = 4375 = 54×7
A20 = 5571 = 32×619
A21 = 6561 = 38    p3
A22 = 8751 = 3×2917    43n1
A23 = 15723 = 32×1747
A24 = 19683 = 39    p3
A25 = 36759 = 3×12253
A26 = 46791 = 33×1733
A27 = 59049 = 310    p3
A28 = 65535 = 3×5×17×257    2n1
A29 = 140103 = 33×5189
A30 = 177147 = 311    p3
A31 = 208191 = 3×29×2393
A32 = 441027 = 32×49003
A33 = 531441 = 312    p3
A34 = 1594323 = 313    p3
A35 = 4190263 = 7×11×54419
A36 = 4782969 = 314    p3
A37 = 9056583 = 33×335429
A38 = 14348907 = 315    p3
A39 = 43046721 = 316    p3
A40 = 57395631 = 3×19131877    43n1
A41 = 129140163 = 317    p3
A42 = 172186887 = 3×57395629    43n1
A43 = 236923383 = 3×1427×55343
A44 = 387420489 = 318    p3
A45 = 918330183 = 33×34012229
A46 = 1162261467 = 319    p3
A47 = 3486784401 = 320    p3
A48 = 3932935775 = 52×29×5424739
A49 = 4294967295 = 3×5×17×257×65537    2n1
A50 = 4764161215 = 5×11×86621113
A51 = 9158284383 = 3×1012×299261
A52 = 10460353203 = 321    p3
A53 = 10774273279 = 7×11×47×509×5849
A54 = 31381059609 = 322    p3

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

Some other sequences are discussed here.

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