

A019279


Superperfect numbers: sigma(sigma(n)) = 2n where sigma is the sumofdivisors function A000203.


61



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

1,1


COMMENTS

Let sigma_m(n) be result of applying sumofdivisors 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^(p1) where 2^p1 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 nth 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


REFERENCES

A. Hoque, H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249253 (2014).


LINKS

Table of n, a(n) for n=1..12.
G. L. Cohen and H. J. J. te Riele, Iterating the sumofdivisors function, Experimental Mathematics, 5 (1996), pp. 93100.
Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 23210869, Volume1, Issue9, November 2013
L. Toth, The alternating sumofdivisors function, 9th Joint Conf. on Math. and Comp. Sci., February 912, 2012, Siofok, Hungary.
L. Toth, A survey of the alternating sumofdivisors function, arXiv:1111.4842, 2011
Eric Weisstein's World of Mathematics, Superperfect Number


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

Select[ 2^Range[60], DivisorSigma[ 1, DivisorSigma[ 1, #]] == 2*# & ] (* JeanFrançois Alcover, Sep 30 2011, assuming powers of 2 *)


PROG

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


CROSSREFS

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.
Sequence in context: A060656 A061286 * A061652 A162119 A213327 A192623
Adjacent sequences: A019276 A019277 A019278 * A019280 A019281 A019282


KEYWORD

nonn,more,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

a(8)a(9) from Jud McCranie, Jun 01 2000
a(10)a(12) from Vincenzo Librandi, Mar 14 2012


STATUS

approved



