A061493 -id:A061493 - 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

A002963 Number of chisel strokes required for Roman numerals for n.

Original entry on oeis.org

1, 2, 3, 3, 2, 3, 4, 5, 3, 2, 3, 4, 5, 5, 4, 5, 6, 7, 5, 4, 5, 6, 7, 7, 6, 7, 8, 9, 7, 6, 7, 8, 9, 9, 8, 9, 10, 11, 9, 4, 5, 6, 7, 7, 6, 7, 8, 9, 7, 2, 3, 4, 5, 5, 4, 5, 6, 7, 5, 4, 5, 6, 7, 7, 6, 7, 8, 9, 7, 6, 7, 8, 9, 9, 8, 9, 10, 11, 9, 8, 9, 10, 11, 11, 10, 11, 12, 13, 11, 4, 5, 6, 7, 7, 6, 7, 8, 9, 7, 2, 3, 4

Offset: 1

N. J. A. Sloane

Chisel strokes for numerals: I,1; V,2; X,2; L,2; C ( = < ),2; D,3; M,4.
For more about Roman numerals, see A006968.
a(A002964(n)) = n. - Reinhard Zumkeller, Apr 14 2013

Nathaniel Johnston, Table of n, a(n) for n = 1..3999
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A119310.

Cf. A061493, A093783.

a002963 = ch 0 . a061493 where
     ch s 0 = s
     ch s x = ch (s + [0,1,2,2,2,2,3,4] !! d') x'
              where  (x',d) = divMod x 10; d' = fromInteger d
-- Reinhard Zumkeller, Apr 14 2013

with(StringTools): A002963 := proc(n) local r: r:=convert(n, roman): return add(parse(SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll(r[j], "I", "1"), "V", "2"), "X", "2"), "L", "2"), "C", "2"), "D", "3"), "M", "4")), j=1..length(r)): end: seq(A002963(n), n=1..102); # Nathaniel Johnston, May 18 2011

a[n_] := Characters[ IntegerString[n, "Roman"]] /. {"I" -> 1, "V" -> 2, "X" -> 2, "L" -> 2, "C" -> 2, "D" -> 3, "M" -> 4} // Total; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Sep 10 2013 *)

{A002963(n, c=[1000, 4, 900, 6, 500, 3, 400, 5, 100, 2, 90, 4, 50, 2, 40, 4, 10, 2, 9, 3, 5, 2, 4, 3, 1, 1], s=0)= forstep(i=1, #c, 2, while(n>=c[i], n-=c[i]; s+=c[i+1])); s} \\ M. F. Hasler, Jul 27 2016

a002963 = lambda n: sum((d+1-(d==2))*(i%5)+(d+2-(d==1))*(i>4) if (i+1)%5 else 3+d+(d==2)*(i==9) for d,i in enumerate(map(int,str(n)[::-1])))
# Nicholas Stefan Georgescu, Feb 27 2023

More terms from David W. Wilson

Data double-checked by M. F. Hasler, Jul 27 2016

A003587 Roman numerals with 1 letter, in numerical order; then those with 2 letters, etc.

Original entry on oeis.org

1, 5, 10, 50, 100, 500, 1000, 2, 4, 6, 9, 11, 15, 20, 40, 51, 55, 60, 90, 101, 105, 110, 150, 200, 400, 501, 505, 510, 550, 600, 900, 1001, 1005, 1010, 1050, 1100, 1500, 2000, 3, 7, 12, 14, 16, 19, 21, 25, 30, 41, 45, 52, 54, 56, 59, 61, 65, 70, 91, 95, 102, 104, 106, 109, 111, 115, 120, 140, 151

Offset: 1

N. J. A. Sloane, J. H. Conway and John Jackson (ab158(AT)freenet.uchsc.edu)

The sequence is finite because 3999 is the largest number that can be written using the symbols I,V,X,L,C,D,M. - J. Lowell, Nov 17 2020

The Romans had symbols for 5000, 10000, 50000, and 100000, but they have no simple equivalents in our current alphabet. See The Book of Numbers, p. 19. The symbol for 100000 is sometimes represented by (((|))). - N. J. A. Sloane, Nov 28 2020

			Written in Roman numerals, the sequence reads: I, V, X, L, C, D, M, II, IV, VI, IX, XI, XV, XX, XL, LI, LV, LX, XC, CI, CV, CX, CL, CC, CD, DI, DV, DX, DL, DC, CM, MI, MV, MX, MV, MC, MD, MM, III, VII, XII, ... - _M. F. Hasler_, Jan 12 2015

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 19.

Maxim Skorohodov, Table of n, a(n) for n = 1..3999

Cf. A003588, A036746, A061493, A006968.

A003587full = SortBy[Range[3999], StringLength[RomanNumeral[#]] &];
A003587full[[;;100]] (* Paolo Xausa, Mar 19 2024 *)

for(d=1,4,for(n=d,d*1000,A006968(n)==d && print1(n","))) \\ M. F. Hasler, Jan 12 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 roman(n):
    m, c, x, i = n//1000, (n%1000)//100, (n%100)//10, n%10
    return "M"*m + f("CDM", c) + f("XLC", x) + f("IVX", i)
def afull():
    return sorted(list(range(1, 4000)), key=lambda x: (len(roman(x)), x))
print(afull()) # Michael S. Branicky, Dec 04 2022

More terms from M. F. Hasler, Jan 12 2015

A036746 Numbers with "long" representations in Roman notation: given by last n letters from ...MMMDCCCLXXXVIII.

Original entry on oeis.org

1, 2, 3, 8, 18, 28, 38, 88, 188, 288, 388, 888, 1888, 2888, 3888

Offset: 1

J. H. Conway

Also: Least number using n symbols when written as Roman numeral, cf. A006968 and A061493. One could prefix a conventional a(0)=0. - M. F. Hasler, Jan 12 2015

Matt Parker, How Roman numerals broke the official dog database, YouTube video (2022).
Gerard Schildberger, The first 3999 numbers in Roman numerals.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A061493, A006968, A003587.

import Data.List (inits, elemIndex)
import Data.Maybe (mapMaybe)
a036746 n = a036746_list !! (n-1)
a036746_list = map (+ 1) $ mapMaybe (`elemIndex` a061493_list)
   (map (read . reverse) $ tail $ inits $ reverse $ show $ a061493 3888)
-- Reinhard Zumkeller, Apr 14 2013

(* assign the 3999 Roman numerals to the variable 'lst' so that lst = {I, II, III, ... MMMCMXCVII, MMMCMXCVIII, MMMCMXCIX}, e.g. using the link to Schildberger's file, then *) f[1] = 1; f[n_] := Block[{k = 2}, While[ StringLength@ SymbolName@ lst[[k]] != n, k++ ]; k]; Array[f, 15] (* Checked by Robert G. Wilson v, Aug 13 2008; edited by M. F. Hasler, Jan 12 2015 *)

A036746(n)=(n%4+8/9)\.1^(n\4) \\ M. F. Hasler, Jan 12 2015

a(n) = min{ k>0 | A006968(k)=n }. - M. F. Hasler, Jan 12 2015

Checked by Robert G. Wilson v, Aug 13 2008

Edited by M. F. Hasler, Jan 12 2015

A078715 Palindromic Roman numerals.

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 20, 30, 50, 100, 190, 200, 300, 500, 1000, 1900, 2000, 3000

Offset: 1

Rick L. Shepherd, Dec 19 2002

This sequence is consistent with the Roman numerals as expressed in the Schildberger link. 4 (usually IV now) could be included in a variant of this sequence as IIII is sometimes used (especially on clock faces). To make this or similar sequences well-defined for numbers larger than 3999, it must be decided whether and how to handle the apostrophus (backward-C), the vinculum (bar), the frame, or even other multiplier notations used at various times in representations of larger Roman numerals. Recalling the "Y2K crisis", will there be a(n even milder) "Y4M crisis"? In particular, is 4000 to be represented as MMMM, (I)(I)(I)(I) (where parentheses are used to represent C and the apostrophus), MV (with vinculum over the V), IV (with vinculum over both I and V) or IIII with vinculum over all four I's? If there is no general agreement, could Roman civilization be at risk (once again)?

Indices of terms in A061493 which are also in A002113. - M. F. Hasler, Jan 12 2015

			I, II, III, V, X, XIX, XX, XXX, L, C, CXC, CC, CCC, D, M, MCM, MM, MMM

Encyclopaedia Britannica, 1981 ed., Vol. 11, "Mathematics, History of", p. 647.
Webster's Third New International Dictionary (Unabridged), 1976 ed., "Cardinal Numbers Table" and footnotes, p. 1549.

Paul Lewis, Roman numerals and calendar
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numerals

Cf. A061493, A006968 (Roman numerals main entry), A002113 (Palindromic Arabic numerals).

Subsequence of A093703.

a078715 n = a078715_list !! (n-1)
a078715_list = filter ((== 1) . a136522 . a061493) [1..3999]
-- Reinhard Zumkeller, Apr 14 2013

Select[Range[3000], PalindromeQ[RomanNumeral[#]] &] (* Paolo Xausa, Mar 03 2024 *)

is_A078715(n)=Vecrev(n=Str(A061493(n)))==Vec(n) \\ M. F. Hasler, Jan 12 2015

A136522(A061493(a(n))) = 1. - Reinhard Zumkeller, Apr 14 2013

A036787 Length of Roman notation for n = length of decimal representation.

Original entry on oeis.org

1, 5, 11, 15, 20, 40, 51, 55, 60, 90, 102, 104, 106, 109, 111, 115, 120, 140, 151, 155, 160, 190, 201, 205, 210, 250, 300, 401, 405, 410, 450, 502, 504, 506, 509, 511, 515, 520, 540, 551, 555, 560, 590, 601, 605, 610, 650, 700, 901, 905, 910, 950, 1003, 1007

Offset: 1

N. J. A. Sloane

			15 = XV has length 2 in both notations.

Nathaniel Johnston, Table of n, a(n) for n = 1..144 (complete up to 3999)

Cf. A036786, A036788.

a036787 n = a036787_list !! (n-1)
a036787_list = [x | x <- [1..], a006968 x == a055642 x]
-- Reinhard Zumkeller, Apr 20 2013

for n from 1 to 3999 do if(length(convert(n, roman)) = length(n))then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011

Select[Range[1100],StringLength[IntegerString[#,"Roman"]] == IntegerLength[ #]&] (* Harvey P. Dale, Jul 25 2011 *)

is_A036787(n)=#Str(n)==#Str(A061493(n)) \\ M. F. Hasler, Jan 12 2015

A006968(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Apr 20 2013

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A057226 Number of different symbols needed to express n as a Roman numeral.

Original entry on oeis.org

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

Offset: 1

Helge T. Blohmer (crusher(AT)k-town.de), Sep 19 2000

Sequence created using classic Roman numerals, i.e., 99 = XCIX, not the questionable IC.

			a(97) = 4 as you need I, V, X and C to express XCVII.

Nathaniel Johnston, Table of n, a(n) for n = 1..3999
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A043537, A061493.

a057226 = a043537 . a061493  -- Reinhard Zumkeller, Apr 14 2013

with(StringTools): A057226 := proc(n) local r: r:=convert(n, roman): return `if`(Search("I",r)>0,1,0) + `if`(Search("V",r)>0,1,0) + `if`(Search("X",r)>0,1,0) + `if`(Search("L",r)>0,1,0) + `if`(Search("C",r)>0,1,0) + `if`(Search("D",r)>0,1,0) + `if`(Search("M",r)>0,1,0): end: seq(A057226(n), n=1..105); # Nathaniel Johnston, May 18 2011

Table[Length[Union[Characters[IntegerString[n,"Roman"]]]],{n,110}] (* Harvey P. Dale, Mar 18 2013 *)

a(n) = A043537(A061493(n)). - Reinhard Zumkeller, Apr 14 2013

A093783 Sum of digits of n in Roman numeral representation.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 8, 11, 10, 11, 12, 13, 16, 15, 16, 17, 18, 21, 20, 21, 22, 23, 26, 25, 26, 27, 28, 31, 30, 31, 32, 33, 36, 35, 36, 37, 38, 41, 60, 61, 62, 63, 66, 65, 66, 67, 68, 71, 50, 51, 52, 53, 56, 55, 56, 57, 58, 61, 60, 61, 62, 63, 66, 65, 66, 67, 68

Offset: 1

Reinhard Zumkeller, May 17 2004

			n=42 == XLII: a(42) = 'X' + 'L' + 'I' + 'I' = 10+50+1+1 = 62.

Nathaniel Johnston, Table of n, a(n) for n = 1..3999
Stephanus Gibbs, Roman Numeral and Date Conversion
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A007953, A057226, A006968.

Cf. A061493, A002963.

a093783 n = q 0 $ a061493 n where
     q s 0 = s
     q s x = q (s + [0,1,5,10,50,100,500,1000] !! d') x'
             where  (x',d) = divMod x 10; d' = fromInteger d
-- Reinhard Zumkeller, Apr 14 2013
(HP 49G calculator)
::
  CK1&Dispatch
  # FF
  ::
    FPTR2 ^DupQIsZero?
    caseSIZEERR
    FPTR2 ^Z>S
    Z0_
    SWAP
    DUPLEN$
    ZERO_DO
    DUP
    ISTOP-INDEX
    SUB$1#
    BINT48
    #-
    BINT4
    OVER#=
    OVER
    BINT9
    #=
    OR
    IT
    #2+
    FPTR2 ^#>Z
    Z10_
    INDEX@
    FPTR2 ^RP#
    FPTR2 ^RMULText
    ROT
    FPTR2 ^RADDext
    SWAPLOOP
    DROP
  ;
;
Gerald Hillier, Sep 08 2015

A093783 := proc(n) local r: r:=convert(n, roman): return add(convert(r[j], arabic), j=1..length(r)): end: seq(A093783(n), n=1..68); # Nathaniel Johnston, May 18 2011

Total[#2 FromRomanNumeral[#1] & @@@ Tally[Characters@ RomanNumeral@ #]] & /@ Range@ 68 (* Michael De Vlieger, Sep 08 2015, Version 10.2 *)

A199921 Number of Roman numerals < 4000 with n letters.

Original entry on oeis.org

7, 31, 93, 215, 389, 573, 691, 691, 573, 389, 215, 93, 31, 7, 1

Offset: 1

Martin Renner, Nov 12 2011

Note that the sequence is completely symmetrical with the addition of the single (notional) Roman string of length zero. - Ian Duff, Jun 27 2017

			a(1) = 7, since there are the seven one-letter roman numerals I, V, X, L, C, D, M.
a(15) = 1, since there is one fifteen-letter roman numeral MMMDCCCLXXXVIII.

Eric Weisstein's World of Mathematics, Roman Numerals.
Wikipedia, Roman numerals

Cf. A003587, A006968.

Cf. A055642, A061493.

import Data.List (group, sort)
a199921 n = a199921_list !! (n-1)
a199921_list = map length $ group $ sort $ map (a055642 . a061493) [1..3999]
-- Reinhard Zumkeller, Apr 14 2013

for i from 1 to 15 do L[i]:={}: od: for n from 1 to 3999 do L[length(convert(n,roman))]:={op(L[length(convert(n,roman))]),n}; od:
seq(nops(L[i]),i=1..15); # Martin Renner, Nov 13 2011

romanLetterCount = Table[0, {15}]; j = 1; While[j < 4000, romanLetterCount[[StringLength[IntegerString[j, "Roman"]]]]++; j++]; romanLetterCount (* Alonso del Arte, Nov 12 2011 *)
Rest[BinCounts[StringLength[RomanNumeral[Range[3999]]]]] (* Paolo Xausa, Mar 19 2024 *)

A003588 Roman numerals with 1 letter, in alphabetical order; then those with 2 letters, etc.

Original entry on oeis.org

100, 500, 1, 50, 1000, 5, 10, 200, 400, 101, 150, 900, 105, 110, 600, 501, 550, 505, 510, 2, 4, 9, 51, 55, 60, 1100, 1500, 1001, 1050, 2000, 1005, 1010, 6, 90, 11, 40, 15, 20, 300, 201, 250, 205, 210, 401, 450, 405, 410, 102, 104, 109, 151, 155, 160, 901, 950, 905, 910, 106

Offset: 1

N. J. A. Sloane, J. H. Conway and John Jackson (ab158(AT)freenet.uchsc.edu)

In this sequence Roman numerals are limited to k <= 3999. - Sean A. Irvine, Dec 04 2022

Maxim Skorohodov, Table of n, a(n) for n = 1..3999

Cf. A003587, A006968, A061493, A036746.

A003588full = FromRomanNumeral[SortBy[RomanNumeral[Range[3999]], StringLength]];
A003588full[[;;100]] (* Paolo Xausa, Mar 19 2024 *)

(Roman(n,s=Vecsmall("IVXLCDM"))=Strchr(apply(c->s[c-48], Vec(Vecsmall(Str(A061493(n))))))); vecsort(vector(4000,n,[#t=Roman(n),t]),,1)[1..100] \\ M. F. Hasler, Jan 12 2015

Corrected, edited and extended by M. F. Hasler, Jan 12 2015

cp's OEIS Frontend

A006968 Number of letters in Roman numeral representation of n.

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A002963 Number of chisel strokes required for Roman numerals for n.

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A003587 Roman numerals with 1 letter, in numerical order; then those with 2 letters, etc.

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A036746 Numbers with "long" representations in Roman notation: given by last n letters from ...MMMDCCCLXXXVIII.

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A078715 Palindromic Roman numerals.

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A036787 Length of Roman notation for n = length of decimal representation.

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