A006968 -id:A006968 - cp's OEIS frontend

A061493 Roman numerals written using 1 for I, 2 for V, 3 for X, 4 for L, 5 for C, 6 for D, 7 for M.

1, 11, 111, 12, 2, 21, 211, 2111, 13, 3, 31, 311, 3111, 312, 32, 321, 3211, 32111, 313, 33, 331, 3311, 33111, 3312, 332, 3321, 33211, 332111, 3313, 333, 3331, 33311, 333111, 33312, 3332, 33321, 333211, 3332111, 33313, 34, 341, 3411, 34111, 3412

Offset: 1

Frank Ellermann, Jun 12 2001

From Daniel Forgues, Jan 16 2015: (Start)
The Romans in the era of the Roman Empire did not have 0 as a number.
The initial "N" (nulla, meaning "none", or nihil, meaning "nothing") was used as a zero symbol in a table of Roman numerals by Bede or his colleague around 725, but even in the Middle Ages it never became a standard. (End) [Corrected by M. F. Hasler and Peter Luschny, Aug 16 2025]
3999 (MMMCMXCIX) is the largest decimal number that has a well-defined Roman numeral representation. Therefore the sequence deliberately stops there to avoid the ambiguous representations of larger numbers. - Jamie Robert Creasey, May 01 2021
The use of 'N' to indicate 0 or "none" long survived in the historic apothecaries' system of measurement, well into the 20th century, to designate quantities in pharmaceutical prescriptions. - M. F. Hasler, Aug 16 2025

			a(14) = 312 because 14 = XIV in Roman, and I,V,X are coded as 1,2,3 respectively.
a(66) = 4321, LXVI is 50+10+5+1 = 66, a(44) = 3412, XLIV is -10+50-1+5 = 44

Reinhard Zumkeller, Table of n, a(n) for n = 1..3999
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals
Wikipedia, 0 (number) in classical antiquity

Cf. A006968, A002963, A003587, A036787, A057226.

Cf. A036746, A036786, A036788, A160676, A160677, A199921.

a061493 n = read $ r 1 [] n :: Integer where
  r _ roms 0 = roms
  r p roms z = case p of
    1 -> r 2 (d '1' '2' '3' m) z'
    2 -> r 3 (d '3' '4' '5' m ++ roms) z'
    3 -> r 4 (d '5' '6' '7' m ++ roms) z'
    4 -> replicate z '7' ++ roms
    where (z',m) = divMod z 10
  d i j k c =
    [[],[i],[i,i],[i,i,i],[i,j],[j],[j,i],[j,i,i],[j,i,i,i],[i,k]] !! c
-- Reinhard Zumkeller, Apr 14 2013

Array[FromDigits[Characters@ RomanNumeral[#] /. {"I" -> 1, "V" -> 2, "X" -> 3, "L" -> 4, "C" -> 5, "D" -> 6, "M" -> 7}] &, 44] (* Michael De Vlieger, May 01 2021 *)

{A061493(n,s="",c=[1000,7,900,57,500,6,400,56,100,5,90,35,50,4,40,34,10,3,9,13,5,2,4,12,1,1])= forstep(i=1,#c,2,while(n>=c[i],n-=c[i];s=Str(s,c[i+1])));eval(s)} \\ 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 int("7"*m + f("567", c) + f("345", x) + f("123", i))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Aug 24 2022

import roman
def A061493(n, d={ord(c):str(i) for i,c in enumerate("NIVXLCDM")}):
    return int(roman.toRoman(n).translate(d)) # M. F. Hasler, Aug 16 2025

a(n)=i <=> A003587(i)=n, for i in {1,...,7}, i.e., A061493 is a left inverse of A003587 on {1,...,7}. - M. F. Hasler, Jan 12 2015

0 removed again by Georg Fischer, Jan 20 2019

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

A105446 Number of symbols in the Roman Fibonacci number representation of n.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 09 2005

The Roman Fibonacci numbers are composed from the values of the Fibonacci Numbers (A000045) with the grammar of the Roman Numerals (A006968) and a few rules to disambiguate.
The alphabet: {1, 2, 3, 5, 8, A=13, B=21, C=34, D=55, E=89, F=144, ...}.
Rule one: of the infinite set of representations of integers by this grammar, always restrict to the subset of those with shortest length.
Rule two: if there are two or more in the subset of shortest representations, restrict to the subset with fewest subtractions [A31 preferred to 188, B31 preferred to 1AA, CA preferred to 8D, DB preferred to AE].
Rule three: if there are two or more representations per Rules one and two, restrict to the subset with the most duplications of characters [22 preferred to 31, 33 preferred to 51, 55 preferred to 82, 88 preferred to A3, BBB preferred to D53, CC preferred to BE]. We do not need a Rule four for a while...
Lemma: no Roman Fibonacci number requires three consecutive instances of the same symbol. Proof: 3*F(i) = F(i+2) + F(i-2).
Question: what is the asymptotic length of the Roman Fibonacci numbers?

			a(1) = 1 because 1 is a Fibonacci number, equal to its own representation as a Roman Fibonacci number.
a(4) = 2 because 4 is not a Fibonacci number, but can be represented as the sum or difference of two Fibonacci numbers, with its Roman Fibonacci number representation being "22" (not "31" per rule three).
a(17) = 3 because the Roman Fibonacci number representation of 17 has three symbols, namely "A22" (not "188" per rule two).
a(80) = 4 because the Roman Fibonacci number representation of 80 has four symbols, namely "DB22".

Cajori, F. A History of Mathematical Notations, 2 vols. Bound as One, Vol. 1: Notations in Elementary Mathematics. New York: Dover, pp. 30-37, 1993.
Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44-45 and 281, 1992.
Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4-5, 1969.

Eric Weisstein's World of Mathematics, Roman Numerals.
Eric Weisstein's World of Mathematics, Fibonacci Numbers.

A105447 = integers with A105446(n) = 2. A105448 = integers with A105446(n) = 3. A105449 = integers with A105446(n) = 4. A105450 = integers with A105446(n) = 5. A023150 = integers with A105446(n) = 6. A105452 = integers with A105446(n) = 7. A105453 = integers with A105446(n) = 8. A105454 = integers with A105446(n) = 9. A105455 = integers with A105446(n) = 10.
Cf. A000045, A006968.
Appears to be a duplicate of A058978.

a(n) = number of symbols in the Roman Fibonacci number representation of n, as defined in "Comments." a(n) = 1 iff n is an element of A000045. a(n) = 2 iff the shortest Roman Fibonacci number representation of n is as the sum or difference of two elements of A000045 and n is not an element of A000045.

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

A036788 Length of Roman notation for n <= length of decimal representation.

Original entry on oeis.org

1, 5, 10, 11, 15, 20, 40, 50, 51, 55, 60, 90, 100, 101, 102, 104, 105, 106, 109, 110, 111, 115, 120, 140, 150, 151, 155, 160, 190, 200, 201, 205, 210, 250, 300, 400, 401, 405, 410, 450, 500, 501, 502, 504, 505, 506, 509, 510, 511, 515, 520, 540, 550, 551, 555

Offset: 1

N. J. A. Sloane

The Roman numeration system used here is the naive one taught in primary school. This sequence, like many others involving numeration systems, is neither well-defined nor interesting for large values of n. - N. J. A. Sloane, Jul 03 2008

			15 = XV has length 2 in both notations.

Nathaniel Johnston, Table of n, a(n) for n = 1..199 (complete up to 3999)
Paul Lewis, Roman numerals

Cf. A036786, A036787, A105269, A119310, A121305.

a036788 n = a036788_list !! (n-1)
a036788_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[560],StringLength[IntegerString[#,"Roman"]]<= IntegerLength[ #]&] (* Harvey P. Dale, Dec 18 2011 *)

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

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

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

A036786 Length of Roman notation for n < length of decimal representation.

Original entry on oeis.org

10, 50, 100, 101, 105, 110, 150, 200, 400, 500, 501, 505, 510, 550, 600, 900, 1000, 1001, 1002, 1004, 1005, 1006, 1009, 1010, 1011, 1015, 1020, 1040, 1050, 1051, 1055, 1060, 1090, 1100, 1101, 1105, 1110, 1150, 1200, 1400, 1500, 1501, 1505, 1510, 1550

Offset: 1

J. H. Conway

			1000 = M is shorter in Roman numerals, so 1000 is in this sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..55 (complete up to 3999)

Cf. A036787, A036788.

a036786 n = a036786_list !! (n-1)
a036786_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[2000],StringLength[IntegerString[#,"Roman"]]Harvey P. Dale, Feb 10 2015 *)

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

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

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

A092196 Number of letters in "old style" Roman numeral representation of n (e.g., IIII rather than IV).

Original entry on oeis.org

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

Offset: 0

Marc LeBrun, Feb 24 2004

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

Also, number of abacus pieces moved (i.e., differing from their initial positions) for the expression of n on a Chinese abacus. A Chinese abacus is also called "suanpan"(CN), "soroban"(JP). - FUNG Cheok Yin, Aug 08 2017

			a(99)=10 because 99 is LXXXXVIIII.

Nathaniel Johnston, Table of n, a(n) for n = 0..3999
Wikipedia, Abacus
Wikipedia, Soroban
Wikipedia, Suanpan
Index entries for sequences related to number of letters in n

Cf. A006968.

A092196 := proc(n) return length(convert(n, roman, period=early)): end: seq(A092196(n),n=1..105); # Nathaniel Johnston, May 18 2011

a(n)=vecsum(apply(v->(v\5)+(v%5), digits(n))); \\ Andrew Howroyd, Oct 19 2017

a(0)=0 prepended by Andrew Howroyd, Oct 19 2017

cp's OEIS Frontend

A061493 Roman numerals written using 1 for I, 2 for V, 3 for X, 4 for L, 5 for C, 6 for D, 7 for M.

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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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A105446 Number of symbols in the Roman Fibonacci number representation of n.

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

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Mathematica

PARI

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A036788 Length of Roman notation for n <= length of decimal representation.

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Haskell

Maple

Mathematica