A057226 -id:A057226 - 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

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 *)

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

A356726 Integers which have in Roman numerals more distinct symbols than any smaller number.

Original entry on oeis.org

1, 4, 14, 44, 144, 444, 1444

Offset: 1

Alain Cousquer, Aug 24 2022

Indices of record highs in A057226.
Smallest number whose Roman notation has exactly n distinct symbols.
The sequence is finite because 1444 is the smallest number using the symbols I,V,X,L,C,D,M.

			For n = 3, a(3) = 14 because 14 = XIV which is the smallest number with 3 symbols in Roman notation.

Cf. A057226, A038378.

kmax=1500; a={}; n=1; For[k=1, k<=kmax, k++, If[Length[DeleteDuplicates[Characters[RomanNumeral[k]]]] == n, AppendTo[a, k]; n++; k=1]]; a (* Stefano Spezia, Aug 26 2022 *)

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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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A093783 Sum of digits of n in Roman numeral representation.

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A356726 Integers which have in Roman numerals more distinct symbols than any smaller number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica