

A284156


4untouchable numbers.


4



298, 1006, 1016, 1108, 1204, 1502, 1940, 2370, 2770, 3358, 3482, 3596, 3688, 3976, 4076, 4164, 4354, 5206, 5634, 5770, 6104, 6206, 6696, 6850, 7008, 7118, 7290, 7496, 7586, 7654, 7812, 7922, 8494, 8546, 8786, 9108, 9122, 9326, 9340, 9476, 9836, 9952, 10318
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OFFSET

1,1


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 kth iterate of s(n). 4untouchable 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 4untouchable numbers have a natural asymptotic density?


LINKS

Anton Mosunov, 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, J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101107.
Paul Pollack, Carl Pomerance, Some problems of Erdos on the sumofdivisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 126.
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, HeeSung Yang, Variant of a theorem of Erdos on the sumofproperdivisors function, Math. Comp., 83 (2014), 19031913.


EXAMPLE

All even numbers less than 298 have a preimage under s4(n), so they are not 4untouchable.
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).
a(2) = 1006, because 1006 = s3(5366) but 5366 is untouchable.
a(3) = 1016, because 1016 = s3(4402) = s3(5378) but both 4402 and 5378 are untouchable.


CROSSREFS

Cf. A005114, A152454, A283152, A284147, A284187.
Sequence in context: A200582 A200581 A200218 * A234665 A234659 A186126
Adjacent sequences: A284153 A284154 A284155 * A284157 A284158 A284159


KEYWORD

nonn


AUTHOR

Anton Mosunov, Mar 21 2017


STATUS

approved



