A003587 -id:A003587 - 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

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

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

A093788 The Roman numerals, with "i" replaced by "1", "v" replaced by "5", "x" replaced by 10, etc.

Original entry on oeis.org

1, 11, 111, 15, 5, 51, 511, 5111, 110, 10, 101, 1011, 10111, 1015, 105, 1051, 10511, 105111, 10110, 1010, 10101, 101011, 1010111, 101015, 10105, 101051, 1010511, 10105111, 1010110, 101010, 1010101, 10101011, 101010111, 10101015, 1010105

Offset: 1

William J. Rapaport (rapaport(AT)buffalo.edu), May 17 2004

A more compact and easier to parse version is A061493, where I, V, X, L, ... are replaced by 1, 2, 3, 4, ... The terms of this sequence can be converted to those of A061493 by changing digits '5' to '2' and deleting each digit '0' upon increasing by 2 the nonzero digit to its left. - M. F. Hasler, Jul 25 2016

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

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

{A093788(n)=A061493(n,,[1000, 1000, 900, 1001000, 500, 500, 400, 100500, 100, 100, 90, 10100, 50, 50, 40, 1050, 10, 10, 9, 110, 5, 5, 4, 15, 1, 1])} \\ M. F. Hasler, Jul 25 2016

Cross-references added and data double-checked by M. F. Hasler, Jul 25 2016

A195526 Numbers written with two Roman numerals.

Original entry on oeis.org

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

Offset: 1

Kausthub Gudipati, Sep 20 2011

The second group of numbers in A003587.

			2=II. 4=IV. 6=VI. 9=IX. 11=XI. 15=XV.

A118639 Smallest number expressible using the next Roman-numeral symbol under the vinculum system.

Original entry on oeis.org

1, 4, 9, 40, 90, 400, 900, 4000, 9000, 40000, 90000, 400000, 900000

Offset: 1

Greg Huber, May 09 2006

			a(1) = I  (first usage of I);
a(2) = IV (first usage of V);
a(3) = IX (first usage of X);
a(4) = XL (first usage of L);
a(5) = XC (first usage of C);
a(6) = CD (first usage of D);
a(7) = CM (first usage of M).

Wikipedia, Roman numerals

Cf. A003587, A118640.

A195602 Primes written with prime number of Roman numerals.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 41, 43, 47, 59, 61, 67, 79, 83, 97, 101, 109, 113, 131, 137, 149, 151, 157, 173, 199, 223, 227, 239, 241, 263, 281, 293, 311, 313, 317, 331, 349, 353, 401, 419, 421, 439, 443, 461, 463, 467, 479, 491, 509, 557, 569, 571, 577, 599, 601, 607, 619, 641, 643, 647

Offset: 1

Juri-Stepan Gerasimov, Sep 21 2011

Cf. A003587, A195526.

Select[Prime[Range[200]],PrimeQ[Length[Characters[IntegerString[#, "Roman"]]]]&] (* Harvey P. Dale, Mar 19 2013 *)

Roman(n)=my(s);while(n,s+=[0,1,2,3,2,1,2,3,4,2][n%10+1];n\=10);s
forprime(p=2,3999,if(isprime(Roman(p)),print1(p", "))) \\ Charles R Greathouse IV, Oct 06 2011

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

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A199921 Number of Roman numerals < 4000 with n letters.

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

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A093788 The Roman numerals, with "i" replaced by "1", "v" replaced by "5", "x" replaced by 10, etc.

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A195526 Numbers written with two Roman numerals.

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A118639 Smallest number expressible using the next Roman-numeral symbol under the vinculum system.

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A195602 Primes written with prime number of Roman numerals.

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