|
|
A283152
|
|
2-untouchable numbers.
|
|
8
|
|
|
208, 250, 362, 396, 412, 428, 438, 452, 478, 486, 494, 508, 672, 712, 716, 772, 844, 900, 906, 950, 1042, 1048, 1086, 1090, 1112, 1132, 1140, 1252, 1262, 1310, 1338, 1372, 1518, 1548, 1574, 1590, 1592, 1644, 1676, 1678, 1686, 1752, 1756, 1796, 1808, 1810, 1854
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|