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A283152 2-untouchable numbers.

%I A283152 #33 Feb 25 2025 13:12:32
%S A283152 208,250,362,396,412,428,438,452,478,486,494,508,672,712,716,772,844,
%T A283152 900,906,950,1042,1048,1086,1090,1112,1132,1140,1252,1262,1310,1338,
%U A283152 1372,1518,1548,1574,1590,1592,1644,1676,1678,1686,1752,1756,1796,1808,1810,1854
%N A283152 2-untouchable numbers.
%C A283152 Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n. 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 A283152 For n > 1, let s2(n) := s(s(n)). 2-untouchable numbers are the numbers that lie in the image of s(n), but not in the image of s2(n). Question: does the set of 2-untouchable numbers have a natural asymptotic density?
%C A283152 Let U(X) denote the total number of 2-untouchable numbers up to X. Then
%C A283152   U(10^4) = 368
%C A283152   U(10^5) = 4143
%C A283152   U(10^6) = 46854
%C A283152   U(10^7) = 508197
%C A283152   U(10^8) = 5348219
%C A283152   U(2*10^8) = 14616451
%H A283152 Anton Mosunov, <a href="/A283152/b283152.txt">Table of n, a(n) for n = 1..10000</a>
%H A283152 K. Chum, R. K. Guy, and M. J. Jacobson, <a href="https://doi.org/10.1080/10586458.2018.1477077">Numerical and statistics analysis of aliquot sequences</a>, Exp. Math. 29 (2020) 414-425, Table 4.
%H A283152 R. K. Guy and 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 A283152 Paul Pollack and 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 A283152 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 A283152 Carl Pomerance and 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 A283152 All even numbers less than 208 have a preimage under s2(n), so they are not 2-untouchable.
%e A283152 a(1) = 208, because 208 = s(268) but 268 is untouchable. Therefore 208 is not in the image of s2(n). Note that 268 is the only preimage of 208 under s(n).
%e A283152 a(2) = 250, because 250 = s(290) but 290 is untouchable.
%e A283152 a(3) = 362, because 362 = s(430) = s(718) but both 430 and 718 are untouchable.
%o A283152 (PARI) preim(n) =  my(v = []); for (k=1, (n-1)^2, if (sigma(k)-k == n, v = concat(v, k))); v;
%o A283152 isunt(n) = if (n==1, 1, for (k=1, (n-1)^2, if (sigma(k)-k == n, return(0))); 1);
%o A283152 isok(n) =  v = preim(n); if (#v, b = 1; for (k=1, #v, b = b && isunt(v[k])); b, 0); \\ _Michel Marcus_, Mar 04 2017
%Y A283152 Cf. A005114, A152454, A284147, A284156, A284187.
%K A283152 nonn
%O A283152 1,1
%A A283152 _Anton Mosunov_, Mar 01 2017