OFFSET
2,3
COMMENTS
All fixed points in base n have at most 5 digits. Proof: In order to be a fixed point, a number with d digits in base n must meet the condition n^d <= d*(n-1)^4, which is only possible for d < 5.
For 5-digit numbers vwxyz in base n, only numbers where v*n^4 + n^3 - 1 <= v^4 + 3*(n-1)^4 or v*n^4 + n^4 - 1 <= v^4 + 4*(n-1)^4 are possible fixed points. v <= 2 for n <= 250.
LINKS
Christian N. K. Anderson, Table of n, a(n) for n = 2..250
EXAMPLE
For a(8)=7, the solutions are {1,16,17,256,257,272,273}. In base 8, these are written as {1, 20, 21, 400, 401, 420, 421}. Because 1^4 = 1, 2^4 + 0^4 = 16, 2^4 + 1^4 = 17, 4^4 + 0^4 + 0^4 = 256, etc., these are the fixed points in base 8.
PROG
(R) inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }
yn=rep(NA, 20)
for(b in 2:20) yn[b]=sum(sapply(1:(1.5*b^4), function(x) sum(inbase(x, b)^4))==1:(1.5*b^4)); yn
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Kevin L. Schwartz and Christian N. K. Anderson, May 24 2013
STATUS
approved