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A088831
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Numbers k whose abundance is 2: sigma(k) - 2k = 2.
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11
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20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504
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OFFSET
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1,1
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COMMENTS
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If 2^k-3 is prime (k is a term of A050414) then 2^(k-1)*(2^k-3) is in the sequence; this fact is a result of the following interesting theorem that I have found. Theorem: If j is an integer and 2^k-(2j+1) is prime then 2^(k-1)*(2^k-(2j+1)) is a solution of the equation sigma(x)=2(x+j). - Farideh Firoozbakht, Feb 23 2005
Note that the fact "if 2^p-1 is prime then 2^(p-1)*(2^p-1) is a perfect number" is also a trivial result of this theorem. All known terms of this sequence are of the form 2^(k-1)*(2^k-3) where 2^k-3 is prime. Conjecture: There are no terms of other forms. So the next terms of this sequence are likely 549754241024, 8796086730752, 140737463189504, 144115187270549504, 2^93*(2^94-3), 2^115*(2^116-3), 2^121*(2^122-3), 2^149*(2^150-3), etc. - Farideh Firoozbakht, Feb 23 2005
The conjecture in the previous comment is incorrect. The first counterexample is 650, which has factorization 2*5^2*13. - T. D. Noe, May 10 2010
Any term x of this sequence can be combined with any term y of A191363 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
If there exists any odd term in this sequence, it must be weird, so it must exceed 10^28. - Alexander Violette, Jan 02 2022
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REFERENCES
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Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.
Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Sec. B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.
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LINKS
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FORMULA
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Solutions to sigma(x)-2*x = 2.
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EXAMPLE
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Abundances of terms in A045768: {-1,2,2,2,2,2,2,2,2,2} so 1 is not here.
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MATHEMATICA
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Select[Range[10^6], DivisorSigma[1, #] - 2 # == 2 &] (* Michael De Vlieger, Feb 25 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Comment and example corrected by T. D. Noe, May 10 2010
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STATUS
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approved
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