A324278 Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 49, 52, 54, 63, 64, 68, 75, 76, 81, 88, 92, 96, 99, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224

Offset: 1

Author

Amiram Eldar, Feb 20 2019

Keywords

• nonn

Comments

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

Crossrefs

Cf. A051377, A126164, A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

Programs

Mathematica

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 Jean-François Alcover at A005114 *)

Extensions

Data corrected by Amiram Eldar, Jan 22 2020