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A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.
(Formerly M5121)
21, 2133, 19521, 176661, 129127041 (list; graph; refs; listen; history; text; internal format)



328256967373616371221, 67585198634817522935331173030319681, and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown. - Jud McCranie, Dec 16 1999

For all k in A014224, 3^(k-1)*(3^k-2) is in this sequence. - M. F. Hasler, Apr 25 2012

The known examples are all of the form 3^(k-1)*(3^k-2), where 3^k-2 is prime (cf. A014224). Conversely, from sigma(3^(k-1)*p)=(3^k-1)/2*(p+1) it is immediate that 2*sigma(n)=3n+1 for such numbers, i.e., they are 2-hyperperfect. (This is "form 3" with p=3 in McCranie's paper.) - M. F. Hasler, Apr 25 2012

a(6) > 10^18. - Max Alekseyev, Nov 25 2019


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. 277-302.

Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, p. 561.

Daniel Minoli, Voice Over MPLS, McGraw-Hill, 2002, New York, NY, see pp. 112-134.

Daniel Minoli and Robert Bear, Hyperperfect Numbers, PME Journal, Fall 1975, pp. 153-157.

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).


Table of n, a(n) for n=1..5.

Antal Bege, Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.

J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

Daniel Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.

Eric Weisstein's World of Mathematics, Hyperperfect Number


t={}; Do[p=3^n-2; If[PrimeQ[p], q=3^(n-1)*p; AppendTo[t, q]], {n, 100}]; t (* Jayanta Basu, May 02 2013 *)


(PARI) is(n)=n==2*(sigma(n)-n-1) + 1; \\ Charles R Greathouse IV, May 01 2013


Cf. A000396, A028499, A028500, A028501, A028502, A034916, A220290.

Sequence in context: A304643 A153848 A221771 * A219104 A219983 A119099

Adjacent sequences:  A007590 A007591 A007592 * A007594 A007595 A007596




N. J. A. Sloane, David W. Wilson



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Last modified July 14 00:36 EDT 2020. Contains 335716 sequences. (Running on oeis4.)