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A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203). 87
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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.
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.
Tomohiro Yamada, On finiteness of odd superperfect numbers, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.
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
EXAMPLE
sigma(sigma(4))=2*4, so 4 is in the sequence.
MATHEMATICA
sigma = DivisorSigma[1, #]&;
For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* Jean-François Alcover, Sep 11 2018 *)
PROG
(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
CROSSREFS
Sequence in context: A271234 A061286 A288756 * A061652 A278913 A306979
KEYWORD
nonn,more,nice
AUTHOR
EXTENSIONS
a(8)-a(9) from Jud McCranie, Jun 01 2000
Corrected by Michel Marcus, Oct 28 2017
STATUS
approved

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Last modified March 19 06:47 EDT 2024. Contains 370953 sequences. (Running on oeis4.)