A088831 Numbers k whose abundance is 2: sigma(k) - 2k = 2.

20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Author

Labos Elemer, Oct 28 2003

Keywords

• nonn
• more

Comments

A subset of A045768.

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

a(11) > 10^12. - Donovan Johnson, Dec 08 2011

a(12) > 10^13. - Giovanni Resta, Mar 29 2013

a(14) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

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

Is there any odd term in this sequence? - Jenaro Tomaszewski, Jan 06 2021

If there exists any odd term in this sequence, it must be weird, so it must exceed 10^28. - Alexander Violette, Jan 02 2022

Examples

			Abundances of terms in A045768: {-1,2,2,2,2,2,2,2,2,2} so 1 is not here.
		

References

• 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.

Links

• P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
• Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.

Crossrefs

Cf. A033880, A045768, A050414, A191363 (deficiency 2).

Programs

Mathematica

Select[Range[10^6], DivisorSigma[1, #] - 2 # == 2 &] (* Michael De Vlieger, Feb 25 2017 *)

PARI

is(n)=sigma(n)==2*n+2 \\ Charles R Greathouse IV, Feb 21 2017

Formula

Solutions to sigma(x)-2*x = 2.

Extensions

One more term from Farideh Firoozbakht, Feb 23 2005

Comment and example corrected by T. D. Noe, May 10 2010

a(10) from Donovan Johnson, Dec 08 2011

a(11) from Giovanni Resta, Mar 29 2013

a(12) from Jud McCranie, Jun 18 2017

a(13) from Hiroaki Yamanouchi, Aug 23 2018