%I M5121 #101 Oct 04 2024 14:18:31

%S 21,2133,19521,176661,129127041

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

%C 328256967373616371221, 67585198634817522935331173030319681, and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown. - _Jud McCranie_, Dec 16 1999

%C For all k in A014224, 3^(k-1)*(3^k-2) is in this sequence. - _M. F. Hasler_, Apr 25 2012

%C 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

%C a(6) > 10^18. - _Max Alekseyev_, Nov 25 2019

%C 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

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, p. 7, Ellipses, Paris 2008.

%D R. K. Guy, Unsolved Problems in Number Theory, B2.

%D 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.

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

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

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

%D Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.

%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Antal Bege, Kinga Fogarasi, <a href="http://www.acta.sapientia.ro/acta-math/C1-1/MATH1-6.PDF">Generalized perfect numbers</a>, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.

%H J. S. McCranie, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html">A study of hyperperfect numbers</a>, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

%H Daniel Minoli, <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0559206-9">Issues In Non-Linear Hyperperfect Numbers</a>, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

%H Daniel Minoli, <a href="http://www.fq.math.ca/Scanned/19-1/minoli.pdf">Structural Issues For Hyperperfect Numbers</a>, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.

%H Tyler Ross, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Ross/ross3.html">A Perfect Number Generalization and Some Euclid-Euler Type Results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperperfectNumber.html">Hyperperfect Number</a>

%t Select[Range[2*10^5], #==2(DivisorSigma[1,#]-#-1)+1 &] (* _Stefano Spezia_, Sep 24 2024 *)

%o (PARI) is(n)=n==2*(sigma(n)-n-1) + 1; \\ _Charles R Greathouse IV_, May 01 2013

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

%K nonn,hard,more

%O 1,1

%A _N. J. A. Sloane_, _David W. Wilson_