A002904 Delete all letters except c, d, i, l, m, v, x from the US English name of n, then read as Roman numeral if possible, otherwise 0.

0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, 55, 1, 0, 1, 9, 5, 1, 1, 0, 0, 0, 0, 0, 4, 9, 5, 1, 1, 1, 1, 1, 1, 1, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 4, 9, 5, 1, 1, 1, 1, 1, 1, 1, 0, 0, 4, 2, 2, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 0, 0, 0, 6, 6, 1, 1, 1, 1, 1, 0, 0, 4, 2, 2, 1, 1, 1, 1, 1, 0, 0, 4, 2, 2, 0

Offset: 1

Author

J. H. Conway

Keywords

• nonn
• word

Comments

From M. F. Hasler, Feb 21 2020: (Start)

There are several standards for Roman numerals, even though experts agree that none of them has ever been universally accepted, cf. references in Wikipedia article.

The current definition of the sequence uses "if possible", so whenever there is a possible interpretation according to any of these standards, it is acceptable.

Non-uniqueness could be caused by rare irregular variants which must be excluded in order to have a well-defined sequence. E.g., iixx will mean 18, not 22, although the standard notation for 18 is xviii (and xiix would also be used sometimes).

If required, the meaning of "possible" must be made more precise (e.g., allowing only "standard subtractive" or additionnally "standard additive" notation; this can be made completely explicit). (End)

There are 1411 nonzero terms; see Branicky link. - Michael S. Branicky, Jan 09 2023

The first term where British English differs from US is a(1001). This gives 500 in US English, but is invalid in British English (1001 is one thousanD anD one). - Paul Duckett, Oct 24 2024

Examples

			To get a(5), write 5 = "five", delete 'f' & 'e' to get "iv", Roman for 4 = a(5).
To get a(35), write 35 = "thirty five", delete all but 'i's and 'v's, to get "iiv", which is not a legal Roman number*, so a(35) = 0 by definition of the sequence. (*It is never allowed to subtract more than one unit 'I' (or 'X' or 'C') from one of the symbols V, L, D representing 5 times a power of 10, as it is never allowed that one of V, L, D is subtracted from one of I, X, C, M.)
		

Links

• Michael S. Branicky, Table of n, a(n) for all nonzero a(n).
• Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video.
• Wikipedia, Roman numerals, as of Feb 19 2020.

Crossrefs

See A092302 and A121305 for other versions.

Formula

From Michael S. Branicky, Jan 09 2023: (Start)

a(100..999) = 0, due to "dd".

a(k*1000) = 500 for k in {1..4, 10, 14, 20..24, 40..44};

a(k*1000+i) = (500 + a(i))*[a(i) != 0] for i in 1..99 for k in {1..4, 10, 14, 20..24, 40..44}; and

a(n) = 0 for all other 1100 <= n <= 999999, due to "id", "vd", "xd" or "dd".

a(n) = 0 for n >= 10^6, due to "ll".

(End)

Extensions

a(35) and beyond from Michael S. Branicky, Jan 09 2023