A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

21, 2133, 19521, 176661, 129127041, 328256967373616371221

Offset: 1

Author

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

Keywords

• nonn
• hard
• more

Comments

67585198634817522935331173030319681 and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown. - Jud McCranie, Dec 16 1999; updated by Max Alekseyev, Jun 03 2025

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

Numbers k for which sigma(k) = (3k+1)/2, thus numbers k such that A000203(k) = A014682(k). Sequence A064989(a(n)), n >= 1, forms a subsequence of A337342. - Antti Karttunen, Aug 26 2020

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. 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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.

Links

• 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.
• 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.
• Eric Weisstein's World of Mathematics, Hyperperfect Number

Crossrefs

Cf. A000203, A000396, A014682, A028499, A028500, A028501, A028502, A034916, A220290, A337342.

Programs

Mathematica

Select[Range[2*10^5], #==2(DivisorSigma[1,#]-#-1)+1 &] (* Stefano Spezia, Sep 24 2024 *)

PARI

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

Extensions

a(6) from Jud McCranie confirmed and added by Max Alekseyev, Jun 03 2025