%I
%S 1,4,8,9,16,20,25,27,28,32,40,44,45,49,52,54,63,64,68,75,76,81,88,92,
%T 96,99,104,108,116,117,121,124,125,128,136,144,147,148,152,153,160,
%U 164,169,171,172,175,176,184,188,189,192,196,200,207,208,212,216,224
%N Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.
%C The terms are conjectural and based on a search for solutions to esigma(x)  x = k for k in the range of the data section and x < 10^12 (esigma(x)  x = A051377(x)  x = A126164(x) is the sum of aliquot exponential divisors of x).  _Amiram Eldar_, Jan 22 2020
%t fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := esigma[n] = Times @@ fun @@@ FactorInteger[n]; untouchableQ[n_] := Catch[ Do[ If[n == esigma[k]k, Throw[True]], {k, 0, (n+1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 1, 130}]][[2, 1]] (* after JeanFrançois Alcover at A005114 *)
%Y Cf. A051377, A126164, A005114, A063948 (unitary), A324276 (biunitary), A324277 (infinitary).
%K nonn
%O 1,2
%A _Amiram Eldar_, Feb 20 2019
%E Data corrected by _Amiram Eldar_, Jan 22 2020
