A284187 5-untouchable numbers.

838, 904, 1970, 2066, 2176, 3134, 3562, 4226, 4756, 5038, 5312, 5580, 5692, 6612, 6706, 7096, 7210, 7384, 9266, 9530, 9704, 10316, 10742, 10828, 11482, 11578, 11724, 12384, 12592, 12682, 13098, 13236, 13772, 14582, 14846, 15184, 15284, 15338, 15484, 15520, 15578

Offset: 1

Author

Anton Mosunov, Mar 21 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). 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?

Examples

			All even numbers less than 838 have a preimage under s5(n), so they are not 5-untouchable.
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).
a(2) = 904, because 904 = s4(4402) = s4(5378) but both 4402 and 5378 are untouchable.
a(3) = 1970, because 1970 = s4(4312) but 4312 is 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.
• 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, A284147, A284156, A363461.

Extensions

Several missing terms inserted by Jinyuan Wang, Jan 07 2025