1,2

Tested for each n and k<200. Otherwise the proof for all k seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.

Tested for each n and k<=1000. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003

The given terms have been tested for all k. - Don Reble, Nov 03, 2003

This is a proper subset of the multiply perfect numbers A007691. E.g. 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.

DivisorSigma[2k-1, n]/n is an integer for all k=1, 2, 3, .., 200, ...

Cf. A066135, A066284, A007691, A066290.

Sequence in context: A054776 A076231 A076234 this_sequence A115678 A048604 A001516

Adjacent sequences: A066286 A066287 A066288 this_sequence A066290 A066291 A066292

nonn

Labos E. (labos(AT)ana.sote.hu), Dec 12 2001

The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003

Extended to 13 confirmed terms by Don Reble, Nov 04, 2003. There is a question whether there are other members below a[13]. However, there are none in Achim's list of multiperfect numbers (see A007691); Rich Schroppel has suggested that that list is complete to 10^70 - if so, a[1..12] are correct; as for a[13], Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble