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   p32n1
A2 = 9 = 32p3
A3 = 15 = 3×5   43n12n1
A4 = 27 = 33p3
A5 = 39 = 3×13   43n1
A6 = 81 = 34p3
A7 = 111 = 3×37   43n1
A8 = 183 = 3×61
A9 = 243 = 35p3
A10 = 255 = 3×5×17   2n1
A11 = 327 = 3×109   43n1
A12 = 363 = 3×112
A13 = 471 = 3×157
A14 = 729 = 36p3
A15 = 2187 = 37p3
A16 = 2199 = 3×733
A17 = 3063 = 3×1021
A18 = 4359 = 3×1453
A19 = 4375 = 54×7
A20 = 5571 = 32×619
A21 = 6561 = 38p3
A22 = 8751 = 3×2917   43n1
A23 = 15723 = 32×1747
A24 = 19683 = 39p3
A25 = 36759 = 3×12253
A26 = 46791 = 33×1733
A27 = 59049 = 310p3
A28 = 65535 = 3×5×17×257   2n1
A29 = 140103 = 33×5189
A30 = 177147 = 311p3
A31 = 208191 = 3×29×2393
A32 = 441027 = 32×49003
A33 = 531441 = 312p3
A34 = 1594323 = 313p3
A35 = 4190263 = 7×11×54419
A36 = 4782969 = 314p3
A37 = 9056583 = 33×335429
A38 = 14348907 = 315p3
A39 = 43046721 = 316p3
A40 = 57395631 = 3×19131877   43n1
A41 = 129140163 = 317p3
A42 = 172186887 = 3×57395629   43n1
A43 = 236923383 = 3×1427×55343
A44 = 387420489 = 318p3
A45 = 918330183 = 33×34012229
A46 = 1162261467 = 319p3
A47 = 3486784401 = 320p3
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 = 321p3
A53 = 10774273279 = 7×11×47×509×5849
A54 = 31381059609 = 322p3


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

Some other sequences are discussed here.



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