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A105417
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Numbers that are pandigital in Roman numerals, using each of the symbols I, V, X, L, C, D and M at least once.
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2
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1444, 1446, 1447, 1448, 1464, 1466, 1467, 1468, 1474, 1476, 1477, 1478, 1484, 1486, 1487, 1488, 1644, 1646, 1647, 1648, 1664, 1666, 1667, 1668, 1674, 1676, 1677, 1678, 1684, 1686, 1687, 1688, 1744, 1746, 1747, 1748, 1764, 1766, 1767, 1768, 1774, 1776
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Contains A105416. The largest member of this sequence is 3888, MMMDCCCLXXXVIII.
There are 192 = 3 * 4^3 pandigital Roman numerals:
3 ways to start: M*, MM*, MMM* (since no Roman numeral stood for 5000)
4 ways for the hundreds: *CD*, *DC*, *DCC*, *DCCC*
4 ways for the decades: *XL*, *LX*, *LXX*, *LXXX*
4 ways for the units: *IV, *VI, *VII, *VIII (End)
Equivalently, numbers of the form abcd with 1 <= a <= 3 and b,c,d in {4,6,7,8}. - M. F. Hasler, Jul 10 2018
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LINKS
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FORMULA
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EXAMPLE
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a(3) = 1447 because its Roman numeral representation, MCDXLVII, uses each of the symbols V, X, L, C, D and M once and the symbol I twice.
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MAPLE
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with(StringTools): for n from 1 to 3999 do r:=convert(n, roman): if Search("I", r) > 0 and Search("V", r) > 0 and Search("X", r) > 0 and Search("L", r) > 0 and Search("C", r) > 0 and Search("D", r) > 0 and Search("M", r) > 0 then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011
A105417 := select(n->nops(convert(convert(n, roman), set))=7, `$`(3888)) # M. F. Hasler, Jul 10 2018
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MATHEMATICA
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Select[Range[900, 1776], ContainsAll[Characters[RomanNumeral[#]], {"M", "C", "D", "X", "L", "I", "V"}]&] (* James C. McMahon, Jan 24 2024 *)
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PROG
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(Haskell)
import Data.List (nub, sort)
a105417 n = a105417_list !! (n-1)
a105417_list = filter ((== "1234567") . sort . nub . show . a061493) [1..3999]
(PARI) A105417=vector(192, i, fromdigits(apply(d->d-!d, digits(i+63, 4)))+555) \\ M. F. Hasler, Jul 10 2018
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CROSSREFS
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KEYWORD
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base,fini,full,easy,nonn
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AUTHOR
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Robert Happelberg (roberthappelberg(AT)yahoo.com), Apr 06 2005
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STATUS
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approved
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