A125142 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.

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, 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, 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

Offset: 1

Author

Yasutoshi Kohmoto, Jan 12 2007, Jan 29 2007

Keywords

• sign

Comments

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.

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

Examples

			SEPSigma^{5}(5)=1, so a(5)=5: 5 -> 4 -> 7 -> 6 -> 2 -> 1
		

Crossrefs

Cf. A126851-A126852, A125140, A125141.

Programs

Maple

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

Extensions

Edited by N. J. A. Sloane at the suggestions of Andrew S. Plewe and R. J. Mathar, May 14 2007, Jun 10 2007

More terms from R. J. Mathar, Jun 07 2007

More terms from R. J. Mathar, Oct 20 2009