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A125142
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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.
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 (mathar(AT)strw.leidenuniv.nl), Jun 07 2007
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EXAMPLE
| SEPSigma^{5}(5)=1, so a(5)=5: 5 -> 4 -> 7 -> 6 -> 2 -> 1
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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 (mathar(AT)strw.leidenuniv.nl), Jun 07 2007
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CROSSREFS
| Cf. A126851-A126852, A125140, A125141.
Sequence in context: A075884 A030750 A059215 * A096352 A197587 A071286
Adjacent sequences: A125139 A125140 A125141 * A125143 A125144 A125145
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KEYWORD
| sign
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AUTHOR
| Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Jan 12 2007, Jan 29 2007
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestions of Andrew Plewe and R. J. Mathar, May 14 2007, Jun 10 2007
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 07 2007
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 20 2009
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