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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).
0
5, 74, 776, 7770, 77794, 778337, 7784712, 77833385, 778307928, 7783494530
OFFSET
1,1
COMMENTS
The value of the asymptotic density of these solutions was asked in the paper by Trudgian.
LINKS
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique (Beograd), Vol. 97, No. 111 (2015), pp. 175-180.
FORMULA
Lim_{n->oo} a(n)/10^n = 0.778...
EXAMPLE
Below 10^1 there are 5 numbers k with usigma(k) > esigma(k): 2, 3, 5, 6, and 7. Thus a(1) = 5.
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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Aug 28 2019
STATUS
approved