[seqfan] Is it possible to find a 5-imperfect number for A127724

michel.marcus at free.fr michel.marcus at free.fr
Mon Jan 8 18:12:10 CET 2018


Hi Seqfans, 




Many 4-imperfect numbers are known. See A127724 a-files. 
The least known instance is 993803899780063855042560, that satisfies x/rho(x) = 4 where rho is A206369. 


But no 5-imperfect is known. 

In an attempt to find a 5-imperfect numbers, I have searched numbers for which denominator(x/rho(x)) is small. 
Here is an extract with x/rho(x) > 4 and denominator(x/rho(x)) <= 5. 







21/5 70024776640986602156981169643610701824000 
17/4 16894666296261085535723520 
13/3 54269201896764616671660406473798293913600000 
22/5 84029788487811863505234925343983738100341800960 
9/2 373316437260251755241798182764378479569038727298776522806597255168000000 
23/5 447425137667270741526846246168650822923647965320869922371356590080000000000 
14/3 23101697828019582727957348094429256309828763084415991060514234912131560924774400000000 
19/4 273159729150 


Only 24/5 and 5/1 are missing. But with denominators <10, actually all 43/9, 24/5, 29/6, 34/7, 39/8, 44/9, 5 are missing. 
That is, quotients above 19/4 are actually missing, a lthough we have the easy 77/16 with 2310, 


How could I improve these results with smaller instances and go higher than 4.75? 


Thanks 
Michel 

























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