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A226353
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Largest integer k in base n whose squared digits sum to sqrt(k).
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2
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1, 49, 169, 36, 1, 1, 2601, 1089, 1, 8836, 33489, 44100, 1, 149769, 128164, 96721, 1, 156816, 1225, 40804, 12321, 831744, 839056, 1149184, 1737124, 3655744, 407044, 1890625, 2208196, 1089, 1, 1466521, 6125625, 2235025, 2832489, 1, 3759721, 6885376, 8844676
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OFFSET
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2,2
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COMMENTS
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Any d-digit number in base n meeting the criterion must also meet the condition d*(n-1)^2 < n^(d/2). Numerically, it can be shown this limits the candidate values to squares < 22*n^4. The larger values are statistically unlikely, and in fact the largest value of k in the first 1000 bases is ~9.96*n^4 in base 775.
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LINKS
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EXAMPLE
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In base 8, the four solutions are the values {1,16,256,2601}, which are written as {1,20,400,5051} in base 8 and
sqrt(1) = 1 = 1^2
sqrt(16) = 4 = 2^2+0^2
sqrt(256) = 16 = 4^2+0^2+0^2
sqrt(2601)= 51 = 5^2+0^2+5^2+1^2
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PROG
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(R) inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }
for(n in 2:50) cat("Base", n, ":", which(sapply((1:(4.7*n^2))^2, function(x) sum(inbase(x, n)^2)==sqrt(x)))^2, "\n")
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CROSSREFS
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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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