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%I #73 Jan 18 2022 06:36:33
%S 20,104,464,650,1952,130304,522752,8382464,134193152,549754241024,
%T 8796086730752,140737463189504,144115187270549504
%N Numbers k whose abundance is 2: sigma(k) - 2k = 2.
%C A subset of A045768.
%C 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
%C 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
%C 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
%C a(11) > 10^12. - _Donovan Johnson_, Dec 08 2011
%C a(12) > 10^13. - _Giovanni Resta_, Mar 29 2013
%C a(14) > 10^18. - _Hiroaki Yamanouchi_, Aug 23 2018
%C 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
%C Is there any odd term in this sequence? - _Jenaro Tomaszewski_, Jan 06 2021
%C If there exists any odd term in this sequence, it must be weird, so it must exceed 10^28. - _Alexander Violette_, Jan 02 2022
%D Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.
%D 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.
%H P. Hagis, and G. L. Cohen, <a href="https://doi.org/10.1017/S1446788700018401">Some Results Concerning Quasiperfect Numbers</a>, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
%F Solutions to sigma(x)-2*x = 2.
%e Abundances of terms in A045768: {-1,2,2,2,2,2,2,2,2,2} so 1 is not here.
%t Select[Range[10^6], DivisorSigma[1, #] - 2 # == 2 &] (* _Michael De Vlieger_, Feb 25 2017 *)
%o (PARI) is(n)=sigma(n)==2*n+2 \\ _Charles R Greathouse IV_, Feb 21 2017
%Y Cf. A033880, A045768, A050414, A191363 (deficiency 2).
%K nonn,more
%O 1,1
%A _Labos Elemer_, Oct 28 2003
%E One more term from _Farideh Firoozbakht_, Feb 23 2005
%E Comment and example corrected by _T. D. Noe_, May 10 2010
%E a(10) from _Donovan Johnson_, Dec 08 2011
%E a(11) from _Giovanni Resta_, Mar 29 2013
%E a(12) from _Jud McCranie_, Jun 18 2017
%E a(13) from _Hiroaki Yamanouchi_, Aug 23 2018
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