A105417 Numbers that are pandigital in Roman numerals, using each of the symbols I, V, X, L, C, D and M at least once.

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

Offset: 1

Author

Robert Happelberg (roberthappelberg(AT)yahoo.com), Apr 06 2005

Keywords

• base
• fini
• full
• easy
• nonn

Comments

Contains A105416. The largest member of this sequence is 3888, MMMDCCCLXXXVIII.

From Daniel Forgues, Jul 31 2011: (Start)

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

Examples

			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.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..192 (full sequence)

Crossrefs

Cf. A061493, A105416.

Programs

Haskell

import Data.List (nub, sort)
a105417 n = a105417_list !! (n-1)
a105417_list = filter ((== "1234567") . sort . nub . show . a061493) [1..3999]
-- Reinhard Zumkeller, Apr 14 2013

Maple

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

Mathematica

Select[Range[900,1776],ContainsAll[Characters[RomanNumeral[#]],{"M","C","D","X","L","I","V"}]&] (* James C. McMahon, Jan 24 2024 *)

PARI

A105417=vector(192,i,fromdigits(apply(d->d-!d,digits(i+63,4)))+555) \\ M. F. Hasler, Jul 10 2018

Formula

A006968(a(n)) >= 7. - Reinhard Zumkeller, Apr 14 2013