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EXAMPLE
| n=6: sigma[2]+sigma[3]+..+sigma[7]=3+4+7+6+12+8=40=sigma[2+3+..+7]= sigma[27]=1+3+9+27=40;quotient=1.
Some smaller values:a(9)=7,a(10)=42,a(11)=48957, a(15)=298,a(19)=97.
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MATHEMATICA
| g[x_, k_] := Apply[Plus, Table[sg[x+j], j, 0, k-1}]]/ sg[Apply[Plus, Table[x+j, j, 0, k-1}]]] Table[fla=1; Do[s=g[n, h]; If[IntegerQ[s]&&Equal[fla, 1], Print[{n, h}]; fla=0], {n, 1, 10000000}], {h, 1, 6}]
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