A126851 -id:A126851 - cp's OEIS frontend

A125141 a(1) = 2; for n>1, a(n)=SENSigma(a(n-1)), where SENSigma(m) = (-1)^((Sum_i r_i)+Omega(m))Sum_{d|m} (-1)^((Sum_j Max(r_j))+Omega(d))d = Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) if 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.

2, 3, 4, 5, 6, 12, 20, 30, 72, 165, 288, 693, 1056, 3024, 9280, 22500, 42845, 60480, 240000, 794580, 1814400, 7040040, 26352000, 98654400, 321552000, 1260230400, 5311834416, 17570520000, 75087810000, 325180275840, 1526817600000

Offset: 1

Yasutoshi Kohmoto, Jan 12 2007

nonn

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.

Cf. A126851, A126852, A125142.

SENSigma := 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: A125141 := proc(nmax) local a ; a := [2] ; while nops(a)< nmax do a := [op(a),SENSigma(op(-1,a))] ; od ; RETURN(a) ; end: A125141(40) ; # R. J. Mathar, May 18 2007

SENSigma[n_] := Module[{Ifs, i, a, r, p }, Ifs = FactorInteger[n]; a = 1; For[i = 1, i <= Length[Ifs], i++, r = Ifs[[i, 2]]; p = Ifs[[i, 1]]; a = a(p(1 - p^r)/(1 - p) - (-1)^r)]; Return[a]];
A125141[nmax_] := Module[{a}, a = {2}; While[Length[a] < nmax, a = Append[a, SENSigma[a[[-1]]]]]; Return[a]];
A125141[40] (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

More terms from R. J. Mathar, May 18 2007

A126852 a(1) = 8; for n>1, a(n) = SPM4Sigma(a(n-1)).

Original entry on oeis.org

8, 15, 12, 14, 18, 33, 20, 42, 36, 77, 60, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Yasutoshi Kohmoto, Feb 24 2007

nonn

After the first 11 terms sequence is constant at 84. - Charles R Greathouse IV, Sep 02 2009

Cf. A126851.

a(n) = A126851(a(n-1)).

Edited by N. J. A. Sloane, May 14 2007

Index in NAME corrected by R. J. Mathar, Mar 13 2024

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.

Original entry on oeis.org

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

Yasutoshi Kohmoto, Jan 12 2007, Jan 29 2007

sign

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

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

Cf. A126851-A126852, A125140, A125141.

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

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

cp's OEIS Frontend

A125141 a(1) = 2; for n>1, a(n)=SENSigma(a(n-1)), where SENSigma(m) = (-1)^((Sum_i r_i)+Omega(m))Sum_{d|m} (-1)^((Sum_j Max(r_j))+Omega(d))d = Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) if 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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A126852 a(1) = 8; for n>1, a(n) = SPM4Sigma(a(n-1)).

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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.

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cp's OEIS Frontend

A125141 a(1) = 2; for n>1, a(n)=SENSigma(a(n-1)), where SENSigma(m) = (-1)^((Sum_i r_i)+Omega(m))*Sum_{d|m} (-1)^((Sum_j Max(r_j))+Omega(d))*d = Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) if 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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Extensions

A126852 a(1) = 8; for n>1, a(n) = SPM4Sigma(a(n-1)).

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A125141 a(1) = 2; for n>1, a(n)=SENSigma(a(n-1)), where SENSigma(m) = (-1)^((Sum_i r_i)+Omega(m))Sum_{d|m} (-1)^((Sum_j Max(r_j))+Omega(d))d = Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) if 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.