A118121 Roman numeral complexity of n.

1, 2, 3, 2, 1, 2, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 4, 3, 2, 3, 4, 5, 4, 3, 4, 5, 5, 4, 3, 4, 5, 5, 5, 4, 4, 5, 5, 5, 2, 3, 4, 5, 4, 3, 4, 5, 6, 4, 1, 2, 3, 4, 3, 2, 3, 4, 5, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 4, 3, 4, 5, 6, 5, 4, 5, 5, 6, 5, 4, 4, 5, 6, 5, 5, 6, 6, 6, 6, 2, 3, 4, 5, 4, 3, 4, 5, 6, 4, 1, 2, 3

Offset: 1

Author

Jonathan Vos Post, May 12 2006

Keywords

• base
• easy
• nonn

Comments

The least number of letters {I, V, X, L, C, D, M} needed to represent n by an expression with conventional Roman numerals, addition, multiplication and parentheses. a(n) <= A006968(n) and a(n) <= A005245(n). Conventional Roman numerals are very efficient at reducing complexity from number of letters in "old style" Roman numerals (A092196) and more primitive representations. In all but two examples shown (38, 88) the use of {+,*} reduces the representation by a single symbol (counting + and *); in these two it saves 2 symbols. In an alternate history, complexity theory and minimum description length could have been invented by Gregorius Catin.

Examples

			a(n) < A006968(n) for these examples. Here "<" means less in letter count:
a(18) = 4 [IX + IX < XVIII]; a(28) = 5 [XIV * II < XXVIII]; a(33) = 5 [XI * III < XXXIII]; a(36) = 4 [VI * VI < XXXVI]; a(37) = 5 [VI * VI + I < XXXVII]; a(38) = 5 [XIX * II < XXXVIII]; a(77) = 5 [XI * VII < LXXVII]; a(78) = 6 [XIII * VI < LXXVIII]; a(81) = 4 [IX * IX < LXXXI]; a(82) = 5 [XLI * II < LXXXII]; a(83) = 6 [XLI * II + I < LXXXIII]; a(84) = 5 [XX * IV < LXXXIV]; a(87) = 6 [IX * IX + VI < LXXXVII]; a(88) = 6 [XI * VIII < LXXXVIII].
		

Links

• Index to sequences related to the complexity of n

Crossrefs

Cf. A005245, A006968, A092196.