A233482 Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.

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

Author

Michel Lagneau, Dec 11 2013

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..333 (terms < 10^11)

Crossrefs

Cf A000005, A000396, A081357, A008472.

Programs

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 *)