OFFSET
2,2
COMMENTS
Values of k in base n have at most 3 digits. Proof: Because sqrt(k) increases faster than the digit sum of k, only numbers with d digits meeting the condition d*(n-1) >= n^(d/2) are candidate fixed points. d < 3 for n > 6, and since there are no fixed points of four or more digits in bases 2 through 5, there are no fixed points in any base with more than 3 digits.
From the above, it can be shown that for three-digit fixed points of the form xyz, x <= 6; also x <= 4 for n > 846. These theoretical upper limits are statistically unlikely, and in fact of the 86356 solutions in bases 2 to 10000, only 6.5% of them begin with 2, and none begin with 3 through 6.
LINKS
Christian N. K. Anderson, Table of n, a(n) for n = 2..10000
EXAMPLE
For a(16)=5 the solutions are the square numbers {1, 36, 100, 225, 441} because in base 16 they are written as {1, 24, 64, E1, 1B9} and
sqrt(1) = 1
sqrt(36) = 6 = 2+4
sqrt(100) = 10 = 6+4
sqrt(225) = 15 = 14+1, and
sqrt(441) = 21 = 1+11+9.
PROG
(R) sapply(2:16, function(n) sum(sapply((1:(n^ifelse(n>6, 1.5, 2)))^2, function(x) sum(inbase(x, n))==sqrt(x))))
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Christian N. K. Anderson and Kevin L. Schwartz, May 25 2013
STATUS
approved