This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 4500 articles have referenced us, often saying "we would not have discovered this result without the OEIS".

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A019279 Superperfect numbers: sigma(sigma(n)) = 2n where sigma is the sum-of-divisors function A000203. 61
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864 (list; graph; refs; listen; history; text; internal format)



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


A. Hoque, H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249-253 (2014).


Table of n, a(n) for n=1..12.

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013

L. Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

L. Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842, 2011

Eric Weisstein's World of Mathematics, Superperfect Number


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


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


Select[ 2^Range[60], DivisorSigma[ 1, DivisorSigma[ 1, #]] == 2*# & ] (* Jean-Fran├žois Alcover, Sep 30 2011, assuming powers of 2 *)


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


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




N. J. A. Sloane.


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

a(10)-a(12) from Vincenzo Librandi, Mar 14 2012



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 24 22:57 EST 2015. Contains 264374 sequences.