A006968 Number of letters in Roman numeral representation of n.

1, 2, 3, 2, 1, 2, 3, 4, 2, 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, 6, 7, 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, 6, 7, 5, 4, 5, 6, 7, 6, 5, 6, 7, 8, 6, 2, 3, 4, 5, 4, 3, 4, 5, 6, 4, 1, 2, 3, 4, 3, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice
• easy
• changed

Comments

How is this sequence defined for large values? - Charles R Greathouse IV, Feb 01 2011

See A078715 for a discussion on the Roman 4M-problem. - Reinhard Zumkeller, Apr 14 2013

The sequence can be considered to be defined via the formula (as A055642 o A061493), so the question is to be posed in A061493, not here. - M. F. Hasler, Jan 12 2015

References

• GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 60.
• Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..3999
• Rec.puzzles, Archive
• Gerard Schildberger, The first 3999 numbers in Roman numerals
• Eric Weisstein's World of Mathematics, Roman Numerals
• Wikipedia, Roman numerals
• Index entries for sequences related to number of letters in n

Crossrefs

Cf. A002963, A036746, A036786, A036787, A036788, A061493, A092196, A160676, A160677, A199921.

Programs

Haskell

a006968 = lenRom 3 where
   lenRom 0 z = z
   lenRom p z = [0, 1, 2, 3, 2, 1, 2, 3, 4, 2] !! m + lenRom (p - 1) z'
                where (z',m) = divMod z 10
-- Reinhard Zumkeller, Apr 14 2013

Maple

A006968 := proc(n) return length(convert(n,roman)): end: seq(A006968(n),n=1..105); # Nathaniel Johnston, May 18 2011

Mathematica

a[n_] := StringLength[ IntegerString[ n, "Roman"]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 27 2011 *)

PARI

A006968(n)=#Str(A061493(n)) \\ M. F. Hasler, Jan 11 2015

Python

def f(s, k):
    return s[:2] if k==4 else (s[1]*(k>=5)+s[0]*(k%5) if k<9 else s[0]+s[2])
def a(n):
    m, c, x, i = n//1000, (n%1000)//100, (n%100)//10, n%10
    return len("M"*m + f("CDM", c) + f("XLC", x) + f("IVX", i))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Mar 03 2024

Python

import roman
def A006968(n): return len(roman.toRoman(n)) # M. F. Hasler, Aug 18 2025

R

nchar(paste(as.roman(1 :1024))) # N. J. A. Sloane, Aug 23 2009, corrected by M. F. Hasler, Aug 18 2025

Formula

A006968 = A055642 o A061493, i.e., a(n) = A055642(A061493(n)). - M. F. Hasler, Jan 11 2015

Extensions

More terms from Eric W. Weisstein