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

Author

Yasutoshi Kohmoto, Jan 12 2007

Keywords

• nonn

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.

Crossrefs

Cf. A126851, A126852, A125142.

Programs

Maple

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

Mathematica

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 *)

Extensions

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