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A092760
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Unitary-sigma unitary-phi perfect numbers.
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3
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6, 20, 72, 272, 2808, 5280, 12480, 65792, 251719680, 4295032832, 39462420480, 2151811200000, 375297105592320, 4238621367336960, 20203489717239782783648394117120, 84353101158454670682576150304666023245622804480
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OFFSET
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1,1
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COMMENTS
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USUP(n) = n/k for some integer k where USUP(n) = A109712(n).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MAPLE
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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
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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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