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%I #15 Oct 28 2021 02:01:37
%S 388,606,696,790,918,1264,1330,1344,1350,1468,1480,1496,1634,1688,
%T 1800,1938,1966,2006,2026,2202,2220,2318,2402,2456,2538,2780,2830,
%U 2916,2962,2966,2998,3224,3544,3806,3926,4208,4292,4330,4404,4446,4466,4468,4608,4632
%N 3-untouchable numbers.
%C Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.
%C Let sk(n) denote the k-th iterate of s(n). 3-untouchable numbers are the numbers that lie in the image of s2(n), but not in the image of s3(n). Question: does the set of 3-untouchable numbers have a natural asymptotic density?
%H Anton Mosunov, <a href="/A284147/b284147.txt">Table of n, a(n) for n = 1..10000</a>
%H Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, <a href="https://arxiv.org/abs/2110.14136">Numerical and Statistical Analysis of Aliquot Sequences</a>, arXiv:2110.14136 [math.NT], 2021.
%H R. K. Guy, J. L. Selfridge, <a href="https://doi.org/10.1090/S0025-5718-1975-0384669-X">What drives an aliquot sequence?</a>, Math. Comp. 29 (129), 1975, 101-107.
%H Paul Pollack, Carl Pomerance, <a href="https://doi.org/10.1090/btran/10">Some problems of Erdos on the sum-of-divisors function</a>, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
%H Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/aliquot.pdf">The first function and its iterates</a>, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.
%H Carl Pomerance, Hee-Sung Yang, <a href="https://doi.org/10.1090/S0025-5718-2013-02775-5">Variant of a theorem of Erdos on the sum-of-proper-divisors function</a>, Math. Comp., 83 (2014), 1903-1913.
%e All even numbers less than 388 have a preimage under s3(n), so they are not 2-untouchable.
%e a(1) = 388, because 388 = s2(668) but 668 is untouchable. Therefore 388 is not in the image of s3(n). Note that 668 is the only preimage of 388 under s2(n).
%e a(2) = 606, because 606 = s2(474) but 474 is untouchable.
%e a(3) = 696, because 696 = s2(276) = s2(306) but both 276 and 306 are untouchable.
%Y Cf. A005114, A152454, A283152, A284156, A284187.
%K nonn
%O 1,1
%A _Anton Mosunov_, Mar 20 2017
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