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COMMENTS
| Is the number of solutions finite? Do solutions to n+kPhi[n]=Sigma[n] exist for all values of k? For k=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 the number of solutions I know below 1000000 is {1, 7, 2, 2, 1, 5, 3, 3, 0, 1, 1}). Not more for larger k.
If 3*2^n-1 is prime for n>0, then 2^n(3*2^n-1) belongs to the sequence; therefore this sequence is infinite if the sequence of primes of the form 3*2^n-1 (A007505) is infinite. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 31 2004
3796=4.13.73 and 60610624=64.199.4759 do not belong to the class of numbers mentioned above by Vandermast.
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MATHEMATICA
| ta={{0}}; k=2; Do[g=n; If[Equal[n+k*EulerPhi[n], DivisorSigma[1, n]], ta=Append[ta, n]; Print[n]], {n, 1, 182000000}]; {ta, g}
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