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Search: id:A058007
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| A058007 |
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Freestyle perfect numbers: let a and b be positive integers, a=(m_i)>1, b=(e_i)>0, (m_1)<(m_2)< .. <(m_k); n = Product_{i=1,2,..,k}(a^b); s = Product_{i=1,2,..,k}(a^(b+1)-1)/(a-1); n such that s-n=n; sequence gives n values. |
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+0
1
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| 6, 28, 60, 84, 90, 120, 336, 496, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8128, 8190, 8280, 9240, 9828, 9900
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Only one odd freestyle perfect number is known: 198585576189, found by Descartes.
Contribution from Daniel Forgues (squid(AT)zensearch.com), Nov 15 2009: (Start)
If we define a spoof-perfect number as follows:
A spoof-perfect number is a number that would be perfect if some (one or more) of its composite factors were wrongly assumed to be prime, i.e. taken as a spoof prime.
then:
Freestyle perfect numbers are the same as the ordered union of perfect numbers and spoof-perfect numbers.
(End)
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B1.
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EXAMPLE
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E.g. n=60=(3^1)*(4^1)*(5^1), s=120=[(3^2-1)*(4^2-1)*(5^2-1)]/[(3-1)*(4-1)*(5-1)] s-n=120-60=n. so 60 is in the sequence.
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CROSSREFS
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Cf. A000396.
Sequence in context: A120624 A138873 A091307 this_sequence A033588 A014635 A034955
Adjacent sequences: A058004 A058005 A058006 this_sequence A058008 A058009 A058010
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KEYWORD
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nonn,new
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AUTHOR
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Naohiro Nomoto (6284968128(AT)geocities.co.jp), Nov 13 2000
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