OFFSET
1,1
COMMENTS
Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n. 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.
For n > 1, let s2(n) := s(s(n)). 2-untouchable numbers are the numbers that lie in the image of s(n), but not in the image of s2(n). Question: does the set of 2-untouchable numbers have a natural asymptotic density?
Let U(X) denote the total number of 2-untouchable numbers up to X. Then
U(10^4) = 368
U(10^5) = 4143
U(10^6) = 46854
U(10^7) = 508197
U(10^8) = 5348219
U(2*10^8) = 14616451
LINKS
Anton Mosunov, Table of n, a(n) for n = 1..10000
K. Chum, R. K. Guy, M. J. Jacobson, Numerical and statistics analysis of aliquot sequences, Exp. Math. 29 (2020) 414-425, Table 4.
R. K. Guy, J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack, 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, Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.
EXAMPLE
All even numbers less than 208 have a preimage under s2(n), so they are not 2-untouchable.
a(1) = 208, because 208 = s(268) but 268 is untouchable. Therefore 208 is not in the image of s2(n). Note that 268 is the only preimage of 208 under s(n).
a(2) = 250, because 250 = s(290) but 290 is untouchable.
a(3) = 362, because 362 = s(430) = s(718) but both 430 and 718 are untouchable.
PROG
(PARI) preim(n) = my(v = []); for (k=1, (n-1)^2, if (sigma(k)-k == n, v = concat(v, k))); v;
isunt(n) = if (n==1, 1, for (k=1, (n-1)^2, if (sigma(k)-k == n, return(0))); 1);
isok(n) = v = preim(n); if (#v, b = 1; for (k=1, #v, b = b && isunt(v[k])); b, 0); \\ Michel Marcus, Mar 04 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Anton Mosunov, Mar 01 2017
STATUS
approved