cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

21, 2133, 19521, 176661, 129127041, 328256967373616371221
Offset: 1

Views

Author

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

Keywords

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

Crossrefs

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

Programs

Extensions

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

Views

Author

Jud McCranie

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Jud McCranie

Keywords

Comments

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

Links

Crossrefs

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

Extensions

a(5) from Max Alekseyev, Nov 18 2019

A028501 18-hyperperfect numbers: n = 18*(sigma(n)-n-1) + 1.

Original entry on oeis.org

1333, 1909, 2469601, 893748277, 322685352001, 8992165119733, 42052982615431201
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

1978419655654505210563957 are also terms. - Donovan Johnson, Nov 20 2012
a(8) > 10^17. - Max Alekseyev, Nov 17 2019

Links

Crossrefs

Cf. A000396, A007593, A220290, A028499, A028500, A028502, A034916.

Extensions

a(5) from Donovan Johnson, Nov 03 2011
a(6)-a(7) from Donovan Johnson confirmed by Max Alekseyev, Nov 21 2019

A100713 Hyperperfect brilliant numbers.

Original entry on oeis.org

21, 697, 1333, 1909, 3901, 96361, 130153, 163201, 2708413, 2768581, 4013833, 4312681, 4658449, 6392257, 7478041, 8766061, 8883841, 9427657, 9699181, 12064333, 14489437, 15042553, 16260901, 16904101, 18116737, 21396313, 28005301, 29751229, 31837801, 36640993
Offset: 1

Views

Author

Jonathan Vos Post, Dec 11 2004

Keywords

Examples

			21 = 3 * 7, 697 = 17 * 41, 1333 = 31 * 43, 1909 = 23 * 83, 3901 = 47 * 83, 96361 = 173 * 557, 130153 = 157 * 829, 163201 = 293 * 557.
a(2) = 697 because 697 is a 12-hyperperfect number, A028500(2) and is a brilliant number because 697 = 17 * 41.

References

  • Richard K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers", Section B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.
  • Joe Roberts, The Lure of the Integers, Washington, DC: Math. Assoc. Amer., p. 177, 1992.

Links

Crossrefs

Cf. A007592, A078972, A001358.

Formula

a(n) is an element in the intersection of A007592 and A078972. a(n)=m(sigma(a(n))-a(n)-1)+1 for some m>1 and a(n) is a semiprime with the same number of digits in each prime factor.

Extensions

More terms from Amiram Eldar, Dec 01 2020
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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