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A226064 Largest fixed point in base n for the sum of the fourth power of its digits. 2
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 (list; graph; refs; listen; history; text; internal format)
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. Using numeric methods, 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

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)