A092196 -id:A092196 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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

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

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

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

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

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

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

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

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

More terms from Eric W. Weisstein

A341737 a(n) is the number of segments necessary to represent n in the Cistercian numeral system.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 18 2021

The sequence is finite because the numeral system of Cistercian monks allows us to represent the numbers from 0 to 9999.

Stefano Spezia, Table of n, a(n) for n = 0..9999
Daniel Carter Beard, The American boys' book of signs, signals and symbols. New York Public Library. Philadelphia: Lippincott. p. 92, 1918.
dCode, Cistercian Monk Numerals
Lauron William De Laurence, The Great Book of Magical Art, Hindu Magic and East Indian Occultism. Chicago: De Laurence Co. p. 174, 1915.
Agrippa von Nettesheim and Heinrich Cornelius, De Occulta Philosophia, pp. CXLII - CXLIII, 1533 (in Latin).
Wikipedia, Cistercian numerals

Cf. A002963, A092196, A119281.

SN:={"0"->1,"1"->2,"2"->2,"3"->2,"4"->2,"5"->3,"6"->2,"7"->3,"8"->3,"9"->4}; OI:={11,15,17,19,22,28,29,51,55,57,59,71,75,77,79,82,88,89,91,92,95,97,98,99}; Table[(Characters[IntegerString[Floor[n/100]]]/. SN//Total)-Length[Characters[IntegerString[Floor[n/100]]]]+1-If[MemberQ[OI,Floor[n/100]],1,0]-Boole[Floor[n/100]==99]+(Characters[IntegerString[Mod[n,100]]]/. SN//Total)-Length[Characters[IntegerString[Mod[n,100]]]]+1-If[MemberQ[OI,Mod[n,100]],1,0]-Boole[Mod[n,100]==99]-1,{n,0,9999}]
(* Function for visualizing a Cistercian numeral *)
CistercianNumeral[n_]:=ResourceFunction["CistercianNumberEncode"][n];

a(n) <= 9.

A092197 Brevity advantage of "new style" over "old style" Roman numerals.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 2, 2, 2, 2, 4, 2, 2, 2, 2, 5, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 6, 0, 0, 0, 0, 2, 0

Offset: 1

Marc LeBrun, Feb 24 2004

			a(4)=2 because old style takes four letters (IIII) versus new style's two (IV).

Nathaniel Johnston, Table of n, a(n) for n = 1..3999

Cf. A006968, A092196.

A092197 := proc(n) return length(convert(n, roman, period=early)) - length(convert(n, roman)): end: seq(A092197(n),n=1..105); # Nathaniel Johnston, May 18 2011

a(n) = A092196(n) - A006968(n).

A118121 Roman numeral complexity of n.

Original entry on oeis.org

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

Jonathan Vos Post, May 12 2006

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.

			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].

Index to sequences related to the complexity of n

Cf. A005245, A006968, A092196.

A119281 Number of counting rods to represent n in the ancient Chinese rod numeral system.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 7, 8, 9, 10, 1, 2, 3

Offset: 0

Rick L. Shepherd, May 12 2006

Contrast with A092196, the number of letters to represent n in ancient Roman numerals. Negative numbers were represented by the same number of rods but usually of a different color (usually black rods with red rods for positive numbers). It's unclear to me whether 0 itself was ever formally considered represented by the absence of all counting rods, but it does seem reasonable that a(0)=0 from the example below.

			a(105) = 6 because 105 was represented on a counting board by placing one counting rod in the compartment for hundreds, no rods where those representing tens were normally placed and five rods in the units compartment.

Wikipedia, Chinese numerals
Wikipedia, Counting rods

Cf. A092196, A007953, A102677.

a(n)= tmp=abs(n); r=0; l=length(Str(tmp)); for(k=1,l, d=tmp-(tmp\10)*10; tmp=tmp\10; if(d<6, r=r+d, r=r+d-4)); r

a(n) = a(-n) = A007953(n) - 4*A102677(n) = A092196(n) + 4*(number of 5s in n).

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A006968 Number of letters in Roman numeral representation of n.

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A341737 a(n) is the number of segments necessary to represent n in the Cistercian numeral system.

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A092197 Brevity advantage of "new style" over "old style" Roman numerals.

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A118121 Roman numeral complexity of n.

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A119281 Number of counting rods to represent n in the ancient Chinese rod numeral system.

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