A226063 Number of fixed points in base n for the sum of the fourth power of its digits.

1, 1, 3, 4, 1, 1, 7, 3, 4, 3, 1, 2, 1, 7, 2, 2, 1, 4, 2, 6, 2, 3, 1, 3, 1, 11, 3, 3, 2, 2, 7, 4, 1, 4, 3, 1, 3, 4, 1, 2, 2, 2, 3, 4, 2, 2, 1, 2, 1, 2, 1, 2, 4, 3, 3, 2, 2, 1, 3, 2, 5, 2, 9, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 5, 1, 5, 5, 4, 2, 5, 3, 2, 2, 3, 3, 1, 2

Offset: 2

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 24 2013

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 2..250

Crossrefs

Cf. A226064 (greatest fixed point).

Cf. A052455 (fixed points in base 10).

Cf. A023052, A046074, A046197.

Programs

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