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A005820
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3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: sum of divisors of n is 3n.
(Formerly M5376)
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22
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OFFSET
| 1,1
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COMMENTS
| Contribution from Daniel Forgues (squid(AT)zensearch.com), May 09 2010: (Start)
Example:
120 = 2^3*3*5
sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4
= (15)*(4)*(6)
= (3*5)*(2^2)*(2*3)
= 2^3*3^2*5
= (3) * (2^3*3*5)
= 3 * 120 (End)
Contribution from Daniel Forgues (squid(AT)zensearch.com), May 11 2010: (120, 672, 523776, 459818240, 1476304896 and 51001180160) are believed to comprise all sous-doubles. - cf. MathWorld. Six 3-perfect numbers are known - cf. The Multiply Perfect Numbers Page. Is there a proof that there are no other 3-perfect numbers?
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REFERENCES
| A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 138.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.
David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.
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LINKS
| Achim Flammenkamp, The Multiply Perfect Numbers Page
Fred Helenius, Link to Glossary and Lists
Walter Nissen, Abundancy : Some Resources
Eric Weisstein's World of Mathematics, Multiperfect Number.
Eric Weisstein's World of Mathematics, Sous-Double.
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EXAMPLE
| sigma(120)=360=3*120
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MATHEMATICA
| AbundantQ[n_]:=DivisorSigma[1, n]==3*n; a={}; Do[If[AbundantQ[n], AppendTo[a, n]], {n, 10^6}]; a [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 07 2008]
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CROSSREFS
| Cf. A007539, A000396, A027687, A046060, A046061.
Sequence in context: A114887 A069085 A039688 * A052787 A052769 A179724
Adjacent sequences: A005817 A005818 A005819 * A005821 A005822 A005823
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KEYWORD
| nonn,nice,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Wells gives the 6th term as 31001180160, but this is an error.
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