|
| |
|
|
A019279
|
|
Superperfect numbers: sigma(sigma(n)) = 2n where sigma is the sum-of-divisors function A000203.
|
|
58
| | |
|
|
|
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.
See also the Cohen-te Reile links under A019276.
The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), 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 (info(AT)polprimos.com), Mar 11 2008
Largest proper divisor of A075398(n) if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Apr 25 2008
|
|
|
REFERENCES
| L. Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842, 2011
|
|
|
LINKS
| Anonymous, Superperfect Numbers:Definition [broken link]
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
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 (info(AT)polprimos.com), Mar 11 2008
Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A075398(n)/2 = A032742(A075398(n)). - Omar E. Pol (info(AT)polprimos.com), Apr 25 2008
|
|
|
EXAMPLE
| sigma(sigma(4))=2*4, so 4 is in the sequence.
|
|
|
MATHEMATICA
| Select[ 2^Range[60], DivisorSigma[ 1, DivisorSigma[ 1, #]] == 2*# & ] (* From Jean-François Alcover, Sep 30 2011, assuming powers of 2 *)
|
|
|
CROSSREFS
| Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A075398.
Sequence in context: A154004 A060656 A061286 * A061652 A162119 A192623
Adjacent sequences: A019276 A019277 A019278 * A019280 A019281 A019282
|
|
|
KEYWORD
| nonn,more,nice,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments and 2 more terms from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Jun 01 2000
|
| |
|
|
