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

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

Offset: 1

Author

Yasutoshi Kohmoto, Mar 20 2004

Keywords

• nonn

Comments

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, ...

Examples

			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, ...
		

Links

• Andrew Lelechenko, Numbers such that A092356(n) = 2*n
• Andrew Lelechenko, Numbers such that A092356(n) = 3*n

Crossrefs

Cf. A091321.

Programs

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

Extensions

Corrected by Andrew Lelechenko, Apr 10 2014