A028501 -id:A028501 - cp's OEIS frontend

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

21, 2133, 19521, 176661, 129127041, 328256967373616371221

Offset: 1

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

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.

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

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

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

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

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

A028499 6-hyperperfect numbers: n = 6*(sigma(n) - n - 1) + 1.

Original entry on oeis.org

301, 16513, 60110701, 1977225901, 2733834545701, 232630479398401, 336823287227717101

Offset: 1

Jud McCranie

(7^k-6)*7^(k-1) is a term for all k in A191469. - Max Alekseyev, Nov 17 2019

Antal Bege, Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.
Eric Weisstein's World of Mathematics, Hyperperfect Number

Cf. A000396, A007593, A028500, A028501, A028502, A034916, A220290.

isok(n) = 6*(sigma(n) - n - 1) + 1 == n; \\ Michel Marcus, Nov 18 2019

a(5) from Donovan Johnson, Nov 20 2012
a(6) from Donovan Johnson confirmed by Max Alekseyev, Nov 17 2019
a(7) from Giovanni Resta confirmed by Max Alekseyev, May 23 2025

A028502 2772-hyperperfect numbers: n = 2772*(sigma(n)-n-1) + 1.

Original entry on oeis.org

95295817, 124035913, 749931337, 4275383113, 47268697363953913

Offset: 1

Jud McCranie

10^19 < a(6) <= 186690534609915040044368953. - Max Alekseyev, Nov 30 2019

Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.

Cf. A000396, A007593, A220290, A028499, A028500, A028501, A034916.

a(5) from Max Alekseyev, Nov 18 2019

A028500 12-hyperperfect numbers: n = 12*(sigma(n) - n - 1) + 1.

Original entry on oeis.org

697, 2041, 1570153, 62722153, 10604156641, 13544168521, 1792155938521

Offset: 1

Jud McCranie

7056410014866537089009269921 is also a term. - Donovan Johnson, Nov 20 2012
Also terms: 13^7*815787979*11621986347871 and 13^23*542800770374370512771595349. - Giovanni Resta, Nov 18 2019
a(8) >= 10^17. - Max Alekseyev, Nov 22 2019

Antal Bege, Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.
Eric Weisstein's World of Mathematics, Hyperperfect Number

Cf. A000396, A007593, A028499, A028501, A028502, A034916, A220290.

isok(n) = 12*(sigma(n) - n - 1) + 1 == n; \\ Michel Marcus, Nov 18 2019

a(7) from Donovan Johnson, Nov 20 2012

cp's OEIS Frontend

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

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A028499 6-hyperperfect numbers: n = 6*(sigma(n) - n - 1) + 1.

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A028502 2772-hyperperfect numbers: n = 2772*(sigma(n)-n-1) + 1.

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A028500 12-hyperperfect numbers: n = 12*(sigma(n) - n - 1) + 1.

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