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Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.
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%I #10 Aug 11 2021 15:30:30

%S 575,2057,2645,3179,4416,8512,12275,33534,94272,138431,203075,218176,

%T 392747,715878,918592,982157,991841,1082176,1205405,1244387,1559616,

%U 1690432,1966912,2344079,2464576,2982976,3386176,3452992,3625792,3821632,3867712,3900497

%N Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.

%C Numbers n such that A000005(n) and A008472(n) are in the sequence A000396. See the sequence A081357 for the sublime numbers.

%H Donovan Johnson, <a href="/A233482/b233482.txt">Table of n, a(n) for n = 1..333</a> (terms < 10^11)

%e 575 is in the sequence because tau(575) = 6 and sopf(575) = 28,

%e 4416 is in the sequence because tau(4416) = 28 and sopf(4416) = 28,

%e 12275 is in the sequence because tau(12275) = 6 and sopf(12275) = 496,

%e 203075 is in the sequence because tau(203075) = 6 and sopf(203075) = 8128.

%p with(numtheory): lst:={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216} :n1:=nops(lst): for n from 1 to 1000000 do :x:=factorset(n):n2:=nops(x): s:=sum('x[i]', 'i'=1..n2):

%p ii:=0:for m from 1 to n1 do:if s=lst[m] then ii:=1:else fi:od:jj:=0:for p from 1 to n1 do:if tau(n)=lst[p] then jj:=1:else fi:od:if ii=1 and jj=1 then printf(`%d, `,n):else fi:od:

%t Select[Range[4*10^6],AllTrue[{DivisorSigma[0,#],Total[FactorInteger[#][[All,1]]]},PerfectNumberQ]&] (* _Harvey P. Dale_, Aug 11 2021 *)

%Y Cf A000005, A000396, A081357, A008472.

%K nonn

%O 1,1

%A _Michel Lagneau_, Dec 11 2013