login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007593 2-hyperperfect numbers: n = 2*(sigma(n)-n-1) + 1.
(Formerly M5121)
10
21, 2133, 19521, 176661, 129127041, 328256967373616371221, 67585198634817522935331173030319681, 443426488243037769923934299701036035201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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) > 4*10^12. - Donovan Johnson, Nov 20 2012

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, pp 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, pp. 561.

Daniel Minoli, Issues in non-linear hyperperfect numbers, Math. Comp., 34 (1980), 639-645.

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).

LINKS

Table of n, a(n) for n=1..8.

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

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A000396, A220290, A028499-A028502, A034916.

Sequence in context: A221122 A153848 A221771 * A219104 A219983 A119099

Adjacent sequences:  A007590 A007591 A007592 * A007594 A007595 A007596

KEYWORD

nonn,hard,more

AUTHOR

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

EXTENSIONS

More terms from Jud McCranie, Dec 16 1999

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 24 19:08 EDT 2014. Contains 240988 sequences.