This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091443 #45 Feb 16 2025 08:32:52 %S A091443 1379454720,14182439040,212517062615531520,27099073228001299660800, %T A091443 680489641226538823680000,15229814702070563916152832000, %U A091443 34111227434420791224041472000,59023729003862626557345792000 %N 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). %C A091443 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. %C A091443 Conjecture: the sequence is finite. %C A091443 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 %H A091443 Sven Simon, <a href="/A091443/b091443.txt">Table of n, a(n) for n = 1..33</a> [Conjectured to be complete] %H A091443 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers Page</a> (See here for the latest information about the search) %H A091443 Achim Flammenkamp, <a href="/A091443/a091443.pdf">he Multiply Perfect Numbers Page</a> [Local copy in pdf format of the page as of Jun 04 2020, made with permission of the author] %H A091443 Achim Flammenkamp, <a href="/A091443/a091443.txt">Datafile_2013_12_31</a> (with 5311 multiperfect numbers). [_Sven Simon_, Feb 15 2014] %H A091443 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect numbers</a> %e A091443 a(1): 1379454720 = 2^8*3*5*7*19*37*73, sopfr(n)= 2^5*5. %Y A091443 Cf. A000203, A001414, A005820, A027687, A046060, A046061. %Y A091443 Intersection of A007691 and A036844. - _Michel Marcus_, Oct 08 2017 %K A091443 nonn %O A091443 1,1 %A A091443 _Sven Simon_, Jan 10 2004