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A233482
Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.
2
575, 2057, 2645, 3179, 4416, 8512, 12275, 33534, 94272, 138431, 203075, 218176, 392747, 715878, 918592, 982157, 991841, 1082176, 1205405, 1244387, 1559616, 1690432, 1966912, 2344079, 2464576, 2982976, 3386176, 3452992, 3625792, 3821632, 3867712, 3900497
OFFSET
1,1
COMMENTS
Numbers n such that A000005(n) and A008472(n) are in the sequence A000396. See the sequence A081357 for the sublime numbers.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..333 (terms < 10^11)
EXAMPLE
575 is in the sequence because tau(575) = 6 and sopf(575) = 28,
4416 is in the sequence because tau(4416) = 28 and sopf(4416) = 28,
12275 is in the sequence because tau(12275) = 6 and sopf(12275) = 496,
203075 is in the sequence because tau(203075) = 6 and sopf(203075) = 8128.
MAPLE
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):
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:
MATHEMATICA
Select[Range[4*10^6], AllTrue[{DivisorSigma[0, #], Total[FactorInteger[#][[All, 1]]]}, PerfectNumberQ]&] (* Harvey P. Dale, Aug 11 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 11 2013
STATUS
approved