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

%I A226063 #21 Jan 03 2022 03:04:41
%S A226063 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,
%T A226063 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,
%U A226063 3,2,2,1,2,5,1,5,5,4,2,5,3,2,2,3,3,1,2
%N A226063 Number of fixed points in base n for the sum of the fourth power of its digits.
%C A226063 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.
%C A226063 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.
%H A226063 Christian N. K. Anderson, <a href="/A226063/b226063.txt">Table of n, a(n) for n = 2..250</a>
%e A226063 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.
%o A226063 (R) inbase=function(n,b) { x=c(); while(n>=b) { x=c(n%%b,x); n=floor(n/b) }; c(n,x) }
%o A226063 yn=rep(NA,20)
%o A226063 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
%Y A226063 Cf. A226064 (greatest fixed point).
%Y A226063 Cf. A052455 (fixed points in base 10).
%Y A226063 Cf. A023052, A046074, A046197.
%K A226063 nonn,base
%O A226063 2,3
%A A226063 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, May 24 2013