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Search: id:A019279
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| A019279 |
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Superperfect numbers: sigma(sigma(n)) = 2n where sigma is the sum-of-divisors function A000203. |
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+0
55
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| 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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
Indices of hexagonal numbers (A000384) that are also even perfect numbers, if there are no odd superperfect numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2008]
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REFERENCES
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G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Anonymous, Superperfect Numbers:Definition
Experimental Mathematics, Home Page
O. E. Pol, Determinacion geometrica de los numeros primos y perfectos".
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FORMULA
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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
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EXAMPLE
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sigma(sigma(4))=2*4, so 4 is in the sequence.
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CROSSREFS
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Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652.
Cf. A032742, A075398.
Cf. A000384. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2008]
Sequence in context: A154004 A060656 A061286 this_sequence A061652 A162119 A155519
Adjacent sequences: A019276 A019277 A019278 this_sequence A019280 A019281 A019282
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KEYWORD
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nonn,more,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments and 2 more terms from Jud McCranie (j.mccranie(AT)comcast.net), Jun 01 2000
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