This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A284147 #24 Jan 06 2025 05:06:41 %S A284147 388,606,696,790,918,1264,1330,1344,1350,1468,1480,1496,1634,1688, %T A284147 1800,1938,1966,1990,2006,2026,2102,2122,2202,2220,2318,2402,2456, %U A284147 2538,2780,2830,2916,2962,2966,2998,3224,3544,3806,3926,4208,4292,4330,4404,4446,4466 %N A284147 3-untouchable numbers. %C A284147 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 A284147 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 A284147 Jinyuan Wang, <a href="/A284147/b284147.txt">Table of n, a(n) for n = 1..10000</a> %H A284147 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. and <a href="https://doi.org/10.1080/10586458.2018.1477077">Exp. Math. 29 (2020) 414-425</a> %H A284147 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 A284147 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 A284147 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 A284147 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 A284147 All even numbers less than 388 have a preimage under s3(n), so they are not 2-untouchable. %e A284147 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 A284147 a(2) = 606, because 606 = s2(474) but 474 is untouchable. %e A284147 a(3) = 696, because 696 = s2(276) = s2(306) but both 276 and 306 are untouchable. %Y A284147 Cf. A005114, A152454, A283152, A284156, A284187. %K A284147 nonn %O A284147 1,1 %A A284147 _Anton Mosunov_, Mar 20 2017 %E A284147 Several missing terms inserted by _Jinyuan Wang_, Jan 05 2025