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A123060
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Least positive integer k such that n has the same number of characters in base k and in Roman numeral representation, or 0 if no such k exists.
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1
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1, 1, 1, 3, 6, 3, 2, 2, 4, 11, 4, 3, 2, 3, 4, 3, 0, 2, 3, 5, 3, 0, 2, 0, 3, 0, 2, 0, 3, 4, 3, 0, 2, 0, 3, 0, 2, 0, 0, 7, 4, 3, 0, 3, 4, 3, 0, 2, 3, 51, 8, 4, 3, 4, 8, 4, 3, 0, 4, 8, 4, 3, 0, 3, 5, 3, 0, 0, 3, 5, 3, 0, 0, 0, 3, 0, 0, 2, 0, 3, 3, 0, 2, 0, 3, 0
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = min{k: StringLength(n base k) = StringLength(Roman(n))}, or 0 if no such k exists. a(n) = min{k: A006968(n) = 1 + floor(log_b(n))}, or 0 if no such k exists.
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EXAMPLE
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a(1) = 1 since Roman(1) = I and 1(base 1) have the same (1) number of characters.
a(4) = 3 since Roman(4) = IV and 11(base 3) have the same (2) number of characters.
a(8) = 2 since Roman(8) = VIII and 1000(base 2) have the same (4) number of characters.
a(10) = 11 since Roman(10) = X and X(base 11) have the same (1) number of characters.
a(11) = 4 since Roman(11) = XI and 23(base 4) have the same (2) number of characters.
a(12) = 3 since Roman(12) = XII and 110(base 3) have the same (3) number of characters.
a(17) = 0 because Roman(17) = XVII has 4 characters, while 17 = 10001(base 2) has 5 characters and 17 = 122(base 3) has 3 characters.
a(30) = 4 because Roman(30) = XXX has 3 characters, as do 110(base 5) and 132(base 4), but Min{4,5} = 4.
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MAPLE
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A123060 := proc(n) local k, l, r: if(n<=3)then return 1:fi: r:=length(convert(n, roman)): for k from 2 to n+1 do l:=nops(convert(n, base, k)): if(l = r)then return k: elif(l<r)then return 0: fi: od: return 0: end: seq(A123060(n), n=1..86); # Nathaniel Johnston, May 18 2011
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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