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 A284156 #21 Jan 11 2025 10:26:20 %S A284156 298,1006,1016,1108,1204,1492,1502,1940,2164,2344,2370,2770,3116,3358, %T A284156 3410,3482,3596,3676,3688,3976,4076,4164,4354,4870,5206,5634,5770, %U A284156 6104,6206,6332,6488,6696,6850,7008,7118,7290,7496,7586,7654,7812,7922,8164,8396,8434 %N A284156 4-untouchable numbers. %C A284156 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 A284156 Let sk(n) denote the k-th iterate of s(n). 4-untouchable numbers are the numbers that lie in the image of s3(n), but not in the image of s4(n). Question: does the set of 4-untouchable numbers have a natural asymptotic density? %H A284156 Jinyuan Wang, <a href="/A284156/b284156.txt">Table of n, a(n) for n = 1..10000</a> %H A284156 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 A284156 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 A284156 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 A284156 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 A284156 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 A284156 All even numbers less than 298 have a preimage under s4(n), so they are not 4-untouchable. %e A284156 a(1) = 298, because 298 = s3(668) but 668 is untouchable. Therefore 298 is not in the image of s4(n). Note that 668 is the only preimage of 298 under s3(n). %e A284156 a(2) = 1006, because 1006 = s3(5366) but 5366 is untouchable. %e A284156 a(3) = 1016, because 1016 = s3(4402) = s3(5378) but both 4402 and 5378 are untouchable. %Y A284156 Cf. A005114, A152454, A283152, A284147, A284187. %K A284156 nonn %O A284156 1,1 %A A284156 _Anton Mosunov_, Mar 21 2017 %E A284156 Several missing terms inserted by _Jinyuan Wang_, Jan 07 2025