|
| |
|
|
A066289
|
|
Numbers n such that Mod[DivisorSigma[2k-1,n],n]=0 holds for all k; i.e. all odd-power-sums of divisors of n are divisible by n.
|
|
4
|
|
|
|
1, 6, 120, 672, 30240, 32760, 31998395520, 796928461056000, 212517062615531520, 680489641226538823680000, 13297004660164711617331200000, 1534736870451951230417633280000, 6070066569710805693016339910206758877366156437562171488352958895095808000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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.
|
|
|
LINKS
|
Table of n, a(n) for n=1..13.
|
|
|
FORMULA
|
DivisorSigma[2k-1, n]/n is an integer for all k=1, 2, 3, .., 200, ...
|
|
|
CROSSREFS
|
Cf. A066135, A066284, A007691, A066290.
Sequence in context: A054776 A076231 A076234 * A170917 A115678 A048604
Adjacent sequences: A066286 A066287 A066288 * A066290 A066291 A066292
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Labos E. (labos(AT)ana.sote.hu), Dec 12 2001
|
|
|
EXTENSIONS
|
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 Schroeppel 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
|
|
|
STATUS
|
approved
|
| |
|
|
