cp's OEIS Frontend

A226064 Largest fixed point in base n for the sum of the fourth power of its digits.

%I A226064 #21 Jan 03 2022 17:00:09
%S A226064 1,1,243,419,1,1,273,1824,9474,10657,1,8194,1,53314,47314,36354,1,
%T A226064 246049,53808,378690,170768,185027,1,247507,1,1002324,722739,278179,
%U A226064 301299,334194,1004643,959859,1,1538803,1798450,1,4168450,2841074,1,1877793,5556355
%N A226064 Largest fixed point in base n for the sum of the fourth power of its digits.
%C A226064 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 A226064 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 A226064 Christian N. K. Anderson, <a href="/A226064/b226064.txt">Table of n, a(n) for n = 2..250</a>
%H A226064 Christian N. K. Anderson, <a href="/A226064/a226064_1.txt">Table of base, largest fixed point, number of fixed points, and a list of all fixed points in base 10 and base n</a> for n = 1..250
%e A226064 The fixed points in base 8 are {1,16,17,256,257,272,273}, because in base 8, these are written as {1,20,21,400,401,420,421} and 1^4 = 1, 2^4 + 0^4 = 16, 2^4 + 1^4 = 17, 4^4 + 0^4 + 0^4 = 256, etc. The largest of these is 273 = a(8).
%o A226064 (R) for(b in 2:50) {
%o A226064     fp=c()
%o A226064     for(w in 1:b-1) for(x in 1:b-1) if((v1=w^4+x^4)<=(v2=w*b^3+x*b^2))
%o A226064         for(y in 1:b-1) if((u1=v1+y^4)<=(u2=v2+y*b) & u1+b^4>u2+b-1) {
%o A226064             z=which(u1+(1:b-1)^4==u2+(1:b-1))-1
%o A226064             if(length(z)) fp=c(fp,u2+z)
%o A226064         }
%o A226064     cat("Base",b,":",fp[-1],"\n")
%o A226064 }
%Y A226064 Cf. A226063 (number of fixed points).
%Y A226064 Cf. A052455 (fixed points in base 10).
%Y A226064 Cf. A023052, A046074, A046197.
%K A226064 nonn,base
%O A226064 2,3
%A A226064 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, May 24 2013