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

1, 1, 243, 419, 1, 1, 273, 1824, 9474, 10657, 1, 8194, 1, 53314, 47314, 36354, 1, 246049, 53808, 378690, 170768, 185027, 1, 247507, 1, 1002324, 722739, 278179, 301299, 334194, 1004643, 959859, 1, 1538803, 1798450, 1, 4168450, 2841074, 1, 1877793, 5556355

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

Christian N. K. Anderson, Table of base, largest fixed point, number of fixed points, and a list of all fixed points in base 10 and base n for n = 1..250

Example

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).

Prog

(R) for(b in 2:50) {
    fp=c()
    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))
        for(y in 1:b-1) if((u1=v1+y^4)<=(u2=v2+y*b) & u1+b^4>u2+b-1) {
            z=which(u1+(1:b-1)^4==u2+(1:b-1))-1
            if(length(z)) fp=c(fp, u2+z)
        }
    cat("Base", b, ":", fp[-1], "\n")
}

Crossrefs

Cf. A226063 (number of fixed points).

Cf. A052455 (fixed points in base 10).

Cf. A023052, A046074, A046197.

Sequence in context: A223021 A046318 A046375 * A157958 A232924 A067838

Adjacent sequences: A226061 A226062 A226063 * A226065 A226066 A226067

Keyword

nonn,base

Author

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

Status

approved