A322446 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).

5, 74, 776, 7770, 77794, 778337, 7784712, 77833385, 778307928, 7783494530

Offset: 1

Author

Amiram Eldar, Aug 28 2019

Keywords

• nonn
• more

Comments

The value of the asymptotic density of these solutions was asked in the paper by Trudgian.

Examples

			Below 10^1 there are 5 numbers k with usigma(k) > esigma(k): 2, 3, 5, 6, and 7. Thus a(1) = 5.
		

Links

• Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique (Beograd), Vol. 97, No. 111 (2015), pp. 175-180.

Crossrefs

Cf. A034448, A051377, A236474.

Programs

Mathematica

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

Formula

Lim_{n->oo} a(n)/10^n = 0.778...