1,1

USUP(n)=n/k for some integer k. If n=Product p_i^r_i then USUP(n)= UnitarySigma(2^r_1)*UnitaryPhi(n/2^r_1) =(2^r_1+1)*Product(p_i^r_i-1), 2<p_i.

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.

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; 2^17*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

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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 15 2008]

Cf. A092788, A091321, A092356

Sequence in context: A146891 A153372 A028402 this_sequence A058494 A147979 A118265

Adjacent sequences: A092757 A092758 A092759 this_sequence A092761 A092762 A092763

nonn

Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp), Apr 14 2004

2808 inserted by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 15 2008