A052455 -id:A052455 - cp's OEIS frontend

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

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

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.

			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).

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

Cf. A226063 (number of fixed points).
Cf. A052455 (fixed points in base 10).
Cf. A023052, A046074, A046197.

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")
}

A383349 Numbers that have the same set of digits as the sum of 4th powers of its digits.

Original entry on oeis.org

0, 1, 488, 668, 686, 848, 866, 884, 1346, 1364, 1436, 1463, 1634, 1643, 2088, 2556, 2565, 2655, 2808, 2880, 3146, 3164, 3416, 3461, 3614, 3641, 4136, 4163, 4316, 4361, 4479, 4497, 4613, 4631, 4749, 4794, 4947, 4974, 5256, 5265, 5526, 5562, 5625, 5652, 6134, 6143

Offset: 1

Jean-Marc Rebert, Apr 24 2025

			488 and 4^4 + 8^4 + 8^4 = 8448 have the same set of digits {4,8}, so 488 is a term.

Cf. A055013, A383328, A383347, A249515, A003132.

Cf. A052455 (a subsequence).

q[k_] := Module[{d = IntegerDigits[k]}, Union[d] == Union[IntegerDigits[Total[d^4]]]]; Select[Range[0, 7000], q] (* Amiram Eldar, Apr 24 2025 *)

isok(k) = my(d=digits(k)); Set(d) == Set(digits(sum(i=1, #d, d[i]^4))); \\ Michel Marcus, Apr 24 2025

def ok(n): return set(s:=str(n)) == set(str(sum(int(d)**4 for d in s)))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Apr 24 2025

cp's OEIS Frontend

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

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