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A092356 UO-sigma multiperfect numbers: n such that A069184(n)/n is an integer. 4

%I #24 Apr 19 2016 01:07:34

%S 1,6,60,1080,6552,36720,47520,87360,222768,288288,8173440,49585536,

%T 203558400,683289600,920387520,4201148160,25486965504,556121548800,

%U 1610457666048,3633511924224,4399770343643136,6075071799091200,9926754576979968,27220195859304960,66800080530869760,629720915643477504

%N UO-sigma multiperfect numbers: n such that A069184(n)/n is an integer.

%C The UO-sigma function is defined by UO-sigma(n) = A069184(n).

%C 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.

%C A UO-sigma perfect number satisfies UO-sigma(n) = k*n for some k.

%C 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.

%C 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, ...

%H Andrew Lelechenko, <a href="/A092356/a092356_2.txt">Numbers such that A092356(n) = 2*n</a>

%H Andrew Lelechenko, <a href="/A092356/a092356_1.txt">Numbers such that A092356(n) = 3*n</a>

%e 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, ...

%o (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

%Y Cf. A091321.

%K nonn

%O 1,2

%A _Yasutoshi Kohmoto_, Mar 20 2004

%E Corrected by _Andrew Lelechenko_, Apr 10 2014

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)