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%I #12 Sep 24 2018 16:53:14
%S 0,1,2,4,5,2,3,6,6,5,6,4,5,3,7,9,10,6,7,7,5,6,7,6,9,5,8,6,7,7,8,11,8,
%T 10,7,-1,-1,7,7,-1,-1,5,6,8,-1,7,8,9,-1,9,12,-1,-1,8,-1,8,-1,7,8,9,10,
%U 8,8,10,10,8,9,12,9,7,8,-1,-1,-1,9,9,10,7,8,12,-1,-1,-1,-1,11,6,9,11,12,-1
%N a(n) = smallest k such that SEPSigma^{k}(n)=1, or -1 if no such k exists. Here SEPSigma(m) = (-1)^(Sum_i r_i)*Sum_{d|m} (-1)^(Sum_j Max(r_j))*d =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^r_i where m=Product_i p_i^r_i, d=Product_j p_j^r_j, p_j^max(r_j) is the largest power of p_j dividing m.
%C By "Max(r_j)" is meant the following: if d|m, d=p^e*q^f, m=p^x*q^y*r^z then Max(e)=x, Max(f)=y.
%C For n=36, no k exists which matches the definition since the iteration reaches a cycle that toggles between 168 and 156 ad infinitum: 36->91->72->169->183->120->104->156->168->156-> etc. In the same fashion, no solutions exist for n=37,40,41,45,49,52,53,... - _R. J. Mathar_, Jun 07 2007
%e SEPSigma^{5}(5)=1, so a(5)=5: 5 -> 4 -> 7 -> 6 -> 2 -> 1
%p A125140 := proc(n) local ifs,i,a,r,p ; ifs := ifactors(n)[2] ; a := 1 ; for i from 1 to nops(ifs) do r := op(2,op(i,ifs)) ; p := op(1,op(i,ifs)) ; a := a*(p*(1-p^r)/(1-p)+(-1)^r) ; od ; RETURN(a) ; end: A125142 := proc(n) local a,nsep; nsep := n ; a :=0 ; while nsep <> 1 do a := a+1 ; nsep := A125140(nsep) ; od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ",A125142(n)) ; od ; # _R. J. Mathar_, Jun 07 2007
%Y Cf. A126851-A126852, A125140, A125141.
%K sign
%O 1,3
%A _Yasutoshi Kohmoto_, Jan 12 2007, Jan 29 2007
%E Edited by _N. J. A. Sloane_ at the suggestions of _Andrew S. Plewe_ and _R. J. Mathar_, May 14 2007, Jun 10 2007
%E More terms from _R. J. Mathar_, Jun 07 2007
%E More terms from _R. J. Mathar_, Oct 20 2009
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