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COMMENTS
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Let sigma^m (j) be the result of applying the sum-of-divisors function (A000203) m times to j; call j (m,k)-perfect if sigma^m (j) = k*j; then this is the sequence of (2,k)-perfect numbers.
For these numbers, the quotient k = sigma(sigma(j))/j is an integer (see A098223). Then also k = (sigma(s)/s)*(sigma(j)/j) with s = sigma(j). That is, k = abundancy(s)*abundancy(j).
So looking at the abundancy of these terms may be interesting. Indeed we see that 459818240 and 51001180160 are actually 3-perfect numbers (A005820), and the reason they are here is that they are coprime to 3. So their sums of divisors are 4-perfect numbers (A027687), yielding q=12.
In a similar way, we can see that the 5-perfect numbers (A046060) that are coprime to 5 will be terms of this sequence with q=30. There are 20 such numbers, the smallest being 13188979363639752997731839211623940096. (End)
It is also interesting to note that for a(2)=8, s=sigma(8)=15 is also a term. This happens to be the case for chains of several terms in a row:
8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;
512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;
29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;
1556480, 3932040, 14008320 with k = 9,13,14;
106151936, 251650560, 955367424 with k = 9,13,14;
312792480, 1505806848 with k = 19,20;
6604416000, 30834059256 with k = 19,20;
9623577600, 46566269568 with k = 19,20.
When j is a term, we can test if s=sigma(j) is also a term; this way we get 6 more terms: 572941926400, 845734196736, 1422976331052, 4010593484800, 11383810648416, 36095341363200.
And the corresponding chains are:
173238912000, 845734196736 with k = 19,20;
355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)
Here are additional chains for the above list:
57120, 217728 with k = 13,14;
343976, 710400 with k = 7,8;
1980342, 5621760 with k = 10,14;
4404480, 14913024 with k = 11,12;
238608384, 775898880 with k = 11,12. (End)
Currently, the coefficient pairs are [1, 1], [3, 4], [4, 7], [7, 8], [8, 9], [9, 10], [9, 13], [10, 14], [11, 12], [13, 14], [16, 17], [16, 21], [17, 18], [19, 20], [23, 24], [25, 26], [25, 31], [27, 28], [29, 30], [31, 32], [32, 33], [37, 38]. It is interesting to note that for some of them, the pair (s,t) also satisfies t=sigma(s). - Michel Marcus, Jul 03 2016; Sep 06 2016
Using these empirical pairs of coefficients in conjunction with the first comment allows us to determine whether some term is the sum of divisors of another yet unknown smaller term. - Michel Marcus, Jul 04 2016
For m in A090748 = A000043 - 1 and c in A205597 (= odd a(n)), c*2^m is in the sequence, unless 2^(m+1)-1 | sigma(c). Indeed, from sigma(x*y) = sigma(x)*sigma(y) for gcd(x,y) = 1, we get sigma(sigma(c*2^m)) = sigma(sigma(c))*2^(m+1), so c*2^m is in the sequence if sigma(sigma(c))/c = k/2 (where k can't be odd: A330598 has no odd c). - M. F. Hasler, Jan 06 2020
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