login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(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. 73
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

LINKS

Table of n, a(n) for n=1..9.

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

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-Fran├žois Alcover, Sep 11 2018 *)

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: A271234 A061286 A288756 * A061652 A278913 A162119

Adjacent sequences:  A019276 A019277 A019278 * A019280 A019281 A019282

KEYWORD

nonn,more,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

Corrected by Michel Marcus, Oct 28 2017

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 23 02:13 EDT 2018. Contains 315271 sequences. (Running on oeis4.)