A284147 3-untouchable numbers.

388, 606, 696, 790, 918, 1264, 1330, 1344, 1350, 1468, 1480, 1496, 1634, 1688, 1800, 1938, 1966, 1990, 2006, 2026, 2102, 2122, 2202, 2220, 2318, 2402, 2456, 2538, 2780, 2830, 2916, 2962, 2966, 2998, 3224, 3544, 3806, 3926, 4208, 4292, 4330, 4404, 4446, 4466

Offset: 1

Author

Anton Mosunov, Mar 20 2017

Keywords

• nonn

Comments

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.

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?

Examples

			All even numbers less than 388 have a preimage under s3(n), so they are not 2-untouchable.
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).
a(2) = 606, because 606 = s2(474) but 474 is untouchable.
a(3) = 696, because 696 = s2(276) = s2(306) but both 276 and 306 are untouchable.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000
• Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, Numerical and Statistical Analysis of Aliquot Sequences, arXiv:2110.14136 [math.NT], 2021. and Exp. Math. 29 (2020) 414-425
• R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
• Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
• Carl Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.
• Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Crossrefs

Cf. A005114, A152454, A283152, A284156, A284187.

Extensions

Several missing terms inserted by Jinyuan Wang, Jan 05 2025