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A019279 Superperfect numbers: sigma(sigma(n)) = 2n where sigma is the sum-of-divisors function A000203. 69
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


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.

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

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 [math.NT], 2011-2014.

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: A271234 A061286 * A061652 A278913 A162119 A213327

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



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Last modified December 11 03:00 EST 2016. Contains 279034 sequences.