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A226064
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Largest fixed point in base n for the sum of the fourth power of its digits.
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2
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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
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OFFSET
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2,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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PROG
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(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")
}
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CROSSREFS
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Cf. A226063 (number of fixed points).
Cf. A052455 (fixed points in base 10).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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