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A092356
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UO-sigma multiperfect numbers: n such that A069184(n)/n is an integer.
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4
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1, 6, 60, 1080, 6552, 36720, 47520, 87360, 222768, 288288, 8173440, 49585536, 203558400, 683289600, 920387520, 4201148160, 25486965504, 556121548800, 1610457666048, 3633511924224, 4399770343643136, 6075071799091200, 9926754576979968, 27220195859304960, 66800080530869760, 629720915643477504
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OFFSET
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1,2
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COMMENTS
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The UO-sigma function is defined by UO-sigma(n) = A069184(n).
E.g., UO-sigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969. So UO-sigma(n) = UnitarySigma(n) if n=2^r, or = sigma(n) if GCD(2,n)=1.
A UO-sigma perfect number satisfies UO-sigma(n) = k*n for some k.
The initial values of k are 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2. However, I conjecture that every positive integer >= 2 must appear.
Some interesting subsequences exist: s(n) := {a(1), a(4), a(9), a(11), ...} has the property that s(n-1)|s(n): 2*3, 2^3*3^2*7*13, 2^5*3^2*7*13*11, 2^7*3^2*7*11*13*43, 2^8*3^2*7*11*13*43*257, ...
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LINKS
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EXAMPLE
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Sequence begins: 2*3, 2^2*3*5, 2^3*3^3*5, 2^3*3^2*7*13, 2^4*3^3*5*17, 2^5*3^3*5*11, 2^6*3*5*7*13, 2^4*3^2*7*13*17, 2^5*3^2*7*13*11, 2^7*3^3*5*11*43, 2^7*3^2*7*11*13*43, ...
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PROG
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(PARI) is(n)=my(e=valuation(n, 2)); (sigma(n>>e) * if(e, 2^e+1, 1)) % n == 0 \\ Charles R Greathouse IV, Apr 10 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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