A092760 Unitary-sigma unitary-phi perfect numbers.

6, 20, 72, 272, 2808, 5280, 12480, 65792, 251719680, 4295032832, 39462420480, 2151811200000, 375297105592320, 4238621367336960, 20203489717239782783648394117120, 84353101158454670682576150304666023245622804480

Offset

1,1

Comments

USUP(n) = n/k for some integer k where USUP(n) = A109712(n).

Links

Table of n, a(n) for n=1..16.

Formula

Numbers of form 2^(2^m)*F_m appear in the sequence, where F_m means Fermat prime 2^(2^m)+1. Because USUP(2^(2^m)*F_m)=UnitarySigma(2^(2^m))*UnitaryPhi(F_m)=(2^(2^m)+1)*(F_m-1)= F_m*2^(2^m)).

Numbers of the following form exist in the sequence. For j=0 to 4, k*product F_i, i=0 to j, F_i means Fermat prime 2^(2^n)+1, k is an integer.

Example

USUP(2^4*7^2)=UnitarySigma(2^4)*UnitaryPhi(7^2)=17*48= 816
So USUP(n) = UnitarySigma(n) if n=2^r = UnitaryPhi(n) if GCD(2,n)=1
Examples : a(1)=2*F_0, a(5)=2^5*11*F_0*F_1, ...., a(12)=2^40*4278255361*F_0*F_1*F_2*F_3*F_4.
Factorizations : 2*3; 2^2*5; 2^3*3^2; 2^4*17; 2^5*3*11*5; 2^6*5*13*3; 2^8*257; 2^12*3*5*17*241; 2^16*65537; 2^14*3*5*7^2*29*113; 2^10*3*5^5*7*11*41*71; 2^17*3*5*17*257*43691; 2^20*3*5*17*257*61681; 2^40*3*5*17*257*65537*4278255361; 2^48*3^6*5*7*11*13*17*23*47*137*193*65537*115903*22253377; 2^48*3^7*5*7*11*13*17*23*47*137*193*1093*65537*115903*22253377

Maple

A047994 := proc(n) local ifs, d ; if n = 1 then 1; else ifs := ifactors(n)[2] ; mul(op(1, op(d, ifs))^op(2, op(d, ifs))-1, d=1..nops(ifs)) ; fi ; end: A006519 := proc(n) local i ; for i in ifactors(n)[2] do if op(1, i) = 2 then RETURN( op(1, i)^op(2, i) ) ; fi ; od: RETURN(1) ; end: Usup := proc(n) local p2 ; p2 := A006519(n) ; (p2+1)*A047994(n/p2) ; end: for n from 1 do if n mod Usup(n) = 0 then print(n) ; fi; od: # R. J. Mathar, Dec 15 2008

Crossrefs

Cf. A092788, A091321, A092356

Sequence in context: A189604 A153372 A028402 * A240043 A058494 A147979

Adjacent sequences: A092757 A092758 A092759 * A092761 A092762 A092763

Keyword

nonn

Author

Yasutoshi Kohmoto, Apr 14 2004

Extensions

2808 inserted by R. J. Mathar, Dec 15 2008

39462420480 and 2151811200000 inserted by Andrew Lelechenko, Apr 10 2014

Status

approved