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 A284187 #21 Feb 25 2025 13:12:36 %S A284187 838,904,1970,2066,2176,3134,3562,4226,4756,5038,5312,5580,5692,6612, %T A284187 6706,7096,7210,7384,9266,9530,9704,10316,10742,10828,11482,11578, %U A284187 11724,12384,12592,12682,13098,13236,13772,14582,14846,15184,15284,15338,15484,15520,15578 %N A284187 5-untouchable numbers. %C A284187 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 A284187 Let sk(n) denote the k-th iterate of s(n). 5-untouchable numbers are the numbers that lie in the image of s4(n), but not in the image of s5(n). Question: does the set of 5-untouchable numbers have a natural asymptotic density? %H A284187 Jinyuan Wang, <a href="/A284187/b284187.txt">Table of n, a(n) for n = 1..10000</a> %H A284187 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 A284187 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 A284187 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 A284187 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 A284187 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 A284187 All even numbers less than 838 have a preimage under s5(n), so they are not 5-untouchable. %e A284187 a(1) = 838, because 838 = s4(2588) but 2588 is untouchable. Therefore 838 is not in the image of s5(n). Note that 2588 is the only preimage of 838 under s4(n). %e A284187 a(2) = 904, because 904 = s4(4402) = s4(5378) but both 4402 and 5378 are untouchable. %e A284187 a(3) = 1970, because 1970 = s4(4312) but 4312 is untouchable. %Y A284187 Cf. A005114, A152454, A283152, A284147, A284156, A363461. %K A284187 nonn %O A284187 1,1 %A A284187 _Anton Mosunov_, Mar 21 2017 %E A284187 Several missing terms inserted by _Jinyuan Wang_, Jan 07 2025