A091443 Multiperfect numbers n which are divisible by sopfr(n) (multiperfect number: sigma(n) = k*n with k integer, sopfr: Sum of prime factors with repetition).

1379454720, 14182439040, 212517062615531520, 27099073228001299660800, 680489641226538823680000, 15229814702070563916152832000, 34111227434420791224041472000, 59023729003862626557345792000

Offset: 1

Author

Sven Simon, Jan 10 2004

Keywords

• nonn

Comments

The sequence contains multiperfect numbers with multiplicity k from 3..8. They are extracted from a list with about 5000 multiperfect numbers with multiplicity from 2..11. Because of the size of these numbers, no numbers with multiplicity k > 8 were found, even though there were about 3000 of them in the list. 95% of the multiperfect numbers with multiplicity from 3..8 are known.

Conjecture: the sequence is finite.

There are 5255 multiperfect numbers known with multiplicity 3 to 11. No more findings for A091443 so we still have 33 multiperfect numbers divisible by their sopfr (without the trivial case 1). With multiplicity 3..8 quite surely all are found (only very few - if any - missing). It is estimated that there are about 2200 with multiplicity 9 and 2091 of them are already found. With multiplicity 10 of estimated 4500 1161 are known. So far no multiperfect number with multiplicity 9 or 10 is divisible by its sopfr (with repetition). Using sopfr without repetition (A114887), there is one number with multiplicity 9 (or more). - Sven Simon, Feb 12 2012

Examples

			a(1): 1379454720 = 2^8*3*5*7*19*37*73, sopfr(n)= 2^5*5.
		

Links

• Sven Simon, Table of n, a(n) for n = 1..33 [Conjectured to be complete]
• Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
• Achim Flammenkamp, he Multiply Perfect Numbers Page [Local copy in pdf format of the page as of Jun 04 2020, made with permission of the author]
• Achim Flammenkamp, Datafile_2013_12_31 (with 5311 multiperfect numbers). [_Sven Simon_, Feb 15 2014]
• Eric Weisstein's World of Mathematics, Multiperfect numbers

Crossrefs

Cf. A000203, A001414, A005820, A027687, A046060, A046061.

Intersection of A007691 and A036844. - Michel Marcus, Oct 08 2017