[seqfan] New "metaperfect" number for A068978

[Allan Wechsler]

The sequence oeis.org/A007429 records the sum, over all divisors of n, of
sigma_1(n).

Sigma_1 itself (oeis.org/A000203) records the sum of the divisors
themselves.

This "nested sigma" calculation causes me to think of A007429 as the
"metasigma" function. Like sigma_1, it is multiplicative. The basis can
easily be seen to be the following:

A007429(p^k) = p^k + 2 p^(k-1) + 3 p^(k-2) + ... + (k+1)

where the coefficient and the exponent always add to k+1.

This "sigma-like" function gives rise to an analog of the multiperfect
numbers, which I think of as "metaperfect". A number N is metaperfect if N
divides A007429(N). These numbers are recorded in oeis.org/A068978. The
entry gives the first 28 examples in the data, found by Benoit Cloitre,
Rick Shepard, and Giovanni Resta. A bit later, Hiroaki  Yamanouchi found
the next three, and recorded them in a B-file.

Yamanouchi also found 168 more examples, for a total of 200, but was not
confident enough of their consecutivity to add them to the B-file; instead,
these 200 metaperfect numbers are listed in their own file.

In the last hour, I found, essentially by hand, an example that Yamanouchi
missed: 930 284 109 364 500, which would fit between Yamanouchi's entries
65 and 66. I'm not sure exactly what to do. Perhaps we should change the
text of the entry so that this file is labeled "Other examples, not
necessarily consecutive", and add my new discovery to it?

I would also appreciate it if somebody could verify the validity of my new
example. Its factorization is 2^2*3^4*5^3*11^2*17*19*73*83*97, and I claim
it is "metaperfect" with order 14. (Because it is not divisible by 7, it
has a "partner" exactly 7 times bigger, which is also metaperfect, but of
order 18. This one is also not in Yamanouchi's list.)