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%I #6 Oct 25 2021 11:11:28
%S 1,6,40,234,588,89376,10805558400
%N Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).
%C The corresponding quotients A160135(k)/k are 1, 1, 1, 1, 1, 2, 3, ...
%C a(8) > 1.5*10^10, if it exists.
%e 6 is a term since its nonexponential divisors are 1, 2 and 3, so A160135(6) = 1 + 2 + 3 = 6 which is divisible by 6.
%e 40 is a term since its nonexponential divisors are 1, 2, 4, 5, 8 and 20, so A160135(40) = 1 + 2 + 4 + 5 + 8 + 20 = 40 which is divisible by 40.
%t esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[1000], Divisible[DivisorSigma[1, #] - esigma[#], #] &]
%Y Cf. A160135.
%Y Similar sequences: A007691, A064594, A064595, A189000, A327158.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Oct 25 2021