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 A092760 Unitary-sigma unitary-phi perfect numbers. 3
 6, 20, 72, 272, 2808, 5280, 12480, 65792, 251719680, 4295032832, 39462420480, 2151811200000, 375297105592320, 4238621367336960, 20203489717239782783648394117120, 84353101158454670682576150304666023245622804480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS USUP(n) = n/k for some integer k where USUP(n) = A109712(n). LINKS 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

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Last modified September 25 07:27 EDT 2021. Contains 347654 sequences. (Running on oeis4.)