OFFSET
1,1
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
Max A. Alekseyev, Computing bounded solutions to linear Diophantine equations with the sum of divisors, arXiv:2601.17832 [math.NT], 2026. See p. 9, Table 2.
Antal Bege and Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
Judson 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
MATHEMATICA
Select[Range[2*10^5], #==2(DivisorSigma[1, #]-#-1)+1 &] (* Stefano Spezia, Sep 24 2024 *)
PROG
(PARI) is(n)=n==2*(sigma(n)-n-1) + 1; \\ Charles R Greathouse IV, May 01 2013
CROSSREFS
KEYWORD
nonn,hard,more,changed
AUTHOR
EXTENSIONS
a(6) from Jud McCranie confirmed and added by Max Alekseyev, Jun 03 2025
STATUS
approved