

A007593


2hyperperfect numbers: n = 2*(sigma(n)  n  1) + 1.
(Formerly M5121)


14




OFFSET

1,1


COMMENTS

328256967373616371221, 67585198634817522935331173030319681, and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown.  Jud McCranie, Dec 16 1999
The known examples are all of the form 3^(k1)*(3^k2), where 3^k2 is prime (cf. A014224). Conversely, from sigma(3^(k1)*p)=(3^k1)/2*(p+1) it is immediate that 2*sigma(n)=3n+1 for such numbers, i.e., they are 2hyperperfect. (This is "form 3" with p=3 in McCranie's paper.)  M. F. Hasler, Apr 25 2012


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 21, p. 7, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277302.
Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, p. 561.
Daniel Minoli, Voice Over MPLS, McGrawHill, 2002, New York, NY, see pp. 112134.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, PME Journal, Fall 1975, pp. 153157.
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


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CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



