A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more
• nice

Comments

Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.

Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24. - Jud McCranie, Jun 01 2000

The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol, Feb 29 2008

The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008

Largest proper divisor of A072868(n) if there are no odd superperfect numbers. - Omar E. Pol, Apr 25 2008

This sequence is a divisibility sequence if there are no odd superperfect numbers. - Charles R Greathouse IV, Mar 14 2012

For n>1, sigma(sigma(a(n))) + phi(phi(a(n))) = (9/4)*a(n). - Farideh Firoozbakht, Mar 02 2015

The term "super perfect number" was coined by Suryanarayana (1969). He and Kanold (1969) gave the general form of even superperfect numbers. - Amiram Eldar, Mar 08 2021

Examples

			sigma(sigma(4))=2*4, so 4 is in the sequence.
		

References

• Dieter Bode, Über eine Verallgemeinerung der vollkommenen Zahlen, Dissertation, Braunschweig, 1971.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B9, pp. 99-100.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, pp. 110-111.
• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, pp. 38-42.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Links

• P. Bundschuh, Aufgabe 601, Elem. Math., Vol. 24 (1969), p. 69; alternative link.
• G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
• G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. Volume 57, Number 2 (1975), 359-364.
• A. Hoque and H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249-253 (2014).
• J. L. Hunsucker and Carl Pomerance, There are no odd superperfect number less than 7*10^24, Indian J. Math., Vol. 17 (1975), pp. 107-120.
• H.-J. Kanold, Über "Super perfect numbers", Elem. Math., Vol. 24 (1969), pp. 61-62; alternative link.
• Graham Lord, Even Perfect and Superperfect Numbers, Elem. Math., Vol. 30 (1975), pp. 87-88.
• H. G. Niederreiter, Solution of Aufgabe 601, Elem. Math., Vol. 25 (1970), pp. 66-67; alternative link.
• Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
• D. Suryanarayana, Super perfect numbers, Elem. Math., Vol. 24 (1969), pp. 16-17; alternative link.
• D. Suryanarayana, There is no superperfect number of the form p^(2*alpha), Elem. Math., Vol. 28 (1973), pp. 148-150; alternative link.
• László Tóth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siófok, Hungary.
• László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
• Eric Weisstein's World of Mathematics, Superperfect Number.
• Wikipedia, Superperfect number.
• Tomohiro Yamada, On finiteness of odd superperfect numbers, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.

Crossrefs

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.

Programs

Mathematica

sigma = DivisorSigma[1, #]&;
For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* Jean-François Alcover, Sep 11 2018 *)

PARI

is(n)=sigma(sigma(n))==2*n \\ Charles R Greathouse IV, Nov 20 2012

Python

from itertools import count, islice
def A019279_gen(): # generator of terms
    return (n for n in count(1) if divisor_sigma(divisor_sigma(n)) == 2*n)
A019279_list = list(islice(A019279_gen(),6)) # Chai Wah Wu, Feb 18 2022

Formula

a(n) = (1 + A000668(n))/2, if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008

Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008

a(n) = 2^A090748(n), if there are no odd superperfect numbers. - Ivan N. Ianakiev, Sep 04 2013

Extensions

a(8)-a(9) from Jud McCranie, Jun 01 2000

Corrected by Michel Marcus, Oct 28 2017