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%I #52 Sep 02 2019 08:13:00
%S 5,74,776,7770,77794,778337,7784712,77833385,778307928,7783494530
%N The number of solutions to usigma(k) > esigma(k) below 10^n, where usigma(k) is the sum of unitary divisors of k (A034448) and esigma(k) is the sum of exponential divisors of k (A051377).
%C The value of the asymptotic density of these solutions was asked in the paper by Trudgian.
%H Tim Trudgian, <a href="https://www.emis.de/journals/PIMB/111/16.html">The sum of the unitary divisor function</a>, Publications de l'Institut Mathématique (Beograd), Vol. 97, No. 111 (2015), pp. 175-180.
%F Lim_{n->oo} a(n)/10^n = 0.778...
%e Below 10^1 there are 5 numbers k with usigma(k) > esigma(k): 2, 3, 5, 6, and 7. Thus a(1) = 5.
%t aQ[1] = False; fun[p_, e_] := DivisorSum[e, p^# &]; aQ[n_] := Times @@ (1 + Power @@@ (f = FactorInteger[n])) > Times @@ (fun @@@ f); c = 0; k = 1; s = {}; Do[While[k < 10^n, If[aQ[k], c++]; k++]; AppendTo[s, c], {n, 1, 6}]; s
%Y Cf. A034448, A051377, A236474.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Aug 28 2019
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