A006968 -id:A006968 - cp's OEIS frontend

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

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

A093703 Numbers whose Roman numeral representation, reversed, is a Roman numeral.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 19, 20, 30, 40, 50, 60, 90, 100, 110, 190, 200, 300, 400, 500, 600, 900, 1000, 1100, 1900, 2000, 3000

Offset: 1

Reinhard Zumkeller, May 17 2004

A subset of this is A078715, palindromic Roman numerals. These are not "old style" Roman numerals (where 4 = IIII).

The sequence contains only values less than 4000, see A078715 for a discussion of the Roman 4M-problem.

			a(1) = 1 because Roman(1) = I and Reversal(I) = I, which is Roman.
a(4) = 4 because Roman(4) = IV and Reversal(IV) = VI, which is Roman.
a(10) = 19 because Roman(19) = XIX which is a palindromic Roman numeral.
a(27) = 900 because Roman(900) = CM and Reversal(CM) = MC, which is Roman.
40 == XL -> LX == 60, therefore 40 and 60 are terms.
1999 is not in the sequence because "MIM" is not a well-formed Roman numeral for 1999, although it looks like one; see Schildberger.

Stephanus Gibbs, Roman Numeral and Date Conversion
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A078715 (palindromic Roman numerals), A061493.

a093703 n = a093703_list !! (n-1)
a093703_list = filter
   ((`elem` map a061493 [1..3999]) . a004086 . a061493) [1..]
-- Reinhard Zumkeller, Apr 14 2013

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

Added sections of text from the erroneous A123054. - N. J. A. Sloane, Apr 15 2013

A131650 Number of symbols in Babylonian numeral representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4

Offset: 1

Alain Van Kerckhoven (alain(AT)avk.org), Sep 10 2007

From Wolfdieter Lang, Feb 21 2017: (Start)
For a(1)..a(59) this sequence coincides with A007953.
For the sexagesimal - decimal representation of n see A055643.
The values of the positions is given in A281863.
The sum of the digits of A055643(n) = a(n).
The number of digits of the representation of n is given in A282622. (End)

Wolfdieter Lang, Table of n, a(n) for n = 1..10000
University of St Andrews, Babylonian numerals.
Wikipedia, Babylonian numerals.
Yale Babylonian Collection, High resolution photographs, descriptions and analysis of the root(2) tablet (YBC 7289).

Cf. A006968, A007953, A055643, A281863, A282622.

More terms from Michel Marcus, Jul 12 2013

Terms a(60)-a(81) from Wolfdieter Lang, Feb 21 2017 (to distinguish it from A007953)

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

A105269 "Long" prime Roman numerals. Smallest prime whose Roman numeral representation has n characters.

Original entry on oeis.org

5, 2, 3, 13, 23, 37, 83, 233, 283, 383, 887, 2383, 2887

Offset: 1

Jonathan Vos Post, Apr 16 2005

Finite because of ambiguity of representation for n > 3999. This is the prime version of the sequence A036746 defined by John H. Conway.

			5=V, 2=II, 3=III, 13 = XIII, 23 = XXIII, 37 = XXXVIII, 83 = LXXXIII, 233 = CCXXXIII, 283 = CLXXXIII, 383 = CCCLXXXIII, 887 = DCCCLXXXVII, 2383 = MMCCCLXXXIII, 2887 = MMDCCCLXXXVII.

Gerard Schildberger, The numbers from 1 to 3999 expressed as Roman numerals.

Cf. A007968, A036746.

a(n) = Min(k in A000040) such that A006968(k) = n.

A105417 Numbers that are pandigital in Roman numerals, using each of the symbols I, V, X, L, C, D and M at least once.

Original entry on oeis.org

1444, 1446, 1447, 1448, 1464, 1466, 1467, 1468, 1474, 1476, 1477, 1478, 1484, 1486, 1487, 1488, 1644, 1646, 1647, 1648, 1664, 1666, 1667, 1668, 1674, 1676, 1677, 1678, 1684, 1686, 1687, 1688, 1744, 1746, 1747, 1748, 1764, 1766, 1767, 1768, 1774, 1776

Offset: 1

Robert Happelberg (roberthappelberg(AT)yahoo.com), Apr 06 2005

Contains A105416. The largest member of this sequence is 3888, MMMDCCCLXXXVIII.
From Daniel Forgues, Jul 31 2011: (Start)
  There are 192 = 3 * 4^3 pandigital Roman numerals:
  3 ways to start: M*, MM*, MMM* (since no Roman numeral stood for 5000)
  4 ways for the hundreds: *CD*, *DC*, *DCC*, *DCCC*
  4 ways for the decades: *XL*, *LX*, *LXX*, *LXXX*
  4 ways for the units: *IV, *VI, *VII, *VIII (End)
Equivalently, numbers of the form abcd with 1 <= a <= 3 and b,c,d in {4,6,7,8}. - M. F. Hasler, Jul 10 2018

			a(3) = 1447 because its Roman numeral representation, MCDXLVII, uses each of the symbols V, X, L, C, D and M once and the symbol I twice.

Nathaniel Johnston, Table of n, a(n) for n = 1..192 (full sequence)

Cf. A061493, A105416.

import Data.List (nub, sort)
a105417 n = a105417_list !! (n-1)
a105417_list = filter ((== "1234567") . sort . nub . show . a061493) [1..3999]
-- Reinhard Zumkeller, Apr 14 2013

with(StringTools): for n from 1 to 3999 do r:=convert(n,roman): if Search("I",r) > 0 and Search("V",r) > 0 and Search("X",r) > 0 and Search("L",r) > 0 and Search("C",r) > 0 and Search("D",r) > 0 and Search("M",r) > 0 then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011
A105417 := select(n->nops(convert(convert(n,roman),set))=7, `$`(3888)) # M. F. Hasler, Jul 10 2018

Select[Range[900,1776],ContainsAll[Characters[RomanNumeral[#]],{"M","C","D","X","L","I","V"}]&] (* James C. McMahon, Jan 24 2024 *)

A105417=vector(192,i,fromdigits(apply(d->d-!d,digits(i+63,4)))+555) \\ M. F. Hasler, Jul 10 2018

A006968(a(n)) >= 7. - Reinhard Zumkeller, Apr 14 2013

A116910 Numbers k such that the number of letters in the Roman numeral for k divides k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 21, 24, 30, 40, 44, 45, 48, 50, 54, 60, 64, 80, 84, 85, 88, 90, 92, 96, 100, 102, 110, 111, 112, 116, 120, 132, 133, 135, 150, 152, 156, 168, 174, 175, 180, 182, 189, 200, 201, 204, 210, 220, 222, 225, 230, 240, 245, 246, 248

Offset: 1

Jonathan Vos Post, Mar 17 2006

			a(23) = 84 because there are 6 letters in LXXXIV and 6 divides 84.
a(44) = 189 because there are 7 letters in CLXXXIX and 7 divides 189.
a(56) = 248 because there are 8 letters in CCXLVIII and 8 divides 248.

Nathaniel Johnston and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 593 terms from Nathaniel Johnston)

Cf. A006968.

for n from 1 to 500 do if(n mod length(convert(n,roman)) = 0)then printf("%d, ",n): fi: od: # Nathaniel Johnston, May 18 2011

Select[Range[250],Divisible[#,StringLength[IntegerString[#,"Roman"]]]&] (* Harvey P. Dale, Feb 13 2013 *)

{k such that A006968(k)|k}.

Corrected by Nathaniel Johnston, May 18 2011

A142958 Roman numerals containing three letters (i.e., III, VII, XII, XIV, ...).

Original entry on oeis.org

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, 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

Offset: 1

David Bodycombe (davidbod(AT)hotmail.com), Jul 12 2008

Nathaniel Johnston, Table of n, a(n) for n = 1..93 (complete up to 3999)
Ron Barnette's Zeno's Coffeehouse, Challenge #56 Result. Mentions this sequence.
Gerard Schildberger, The first 3999 numbers in Roman numerals.

Cf. A178968.

for n from 1 to 3999 do if(length(convert(n, roman)) = 3)then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011

(* go to the hyperlink and assign the first 700 Roman numerals to the variable 'lst' so that lst = {I, II, III, ..., DCXCVIII, DCXCIX, DCC} and then *) Select[ Range[2, 1000], StringLength@ SymbolName@ lst[[ # ]] == 3 &] (* Robert G. Wilson v, Aug 12 2008 *)

More terms from Robert G. Wilson v, Aug 12 2008

A222581 Run lengths of digits when concatenating Roman numerals less than 4000, cf. A093796.

Original entry on oeis.org

7, 3, 1, 1, 2, 1, 4, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 7, 1, 3, 2, 3, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 2, 3, 1, 3, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 14 2013

See A078715 for a discussion on the Roman 4M-problem;

a(n) <= 7, that is, the longest run of consecutive equal digits in A093796 has length = 7; see also example.

			The 3999 Roman numerals of all numbers less than 4000 consist of 30000 digits; there are 19770 runs of consecutive equal digits: a(19770) = 1 is the last term of this sequence;
a(1)=a(52)=7, there are two runs with length 7: the first is "IIIIIII" which is the prefix of the concatenation of I, II, III and IV, the second is "XXXXXXX" which is contained in the concatenation of XXIX, XXX and XXXI;
a(1022)=a(14573)=6, there are also two runs with length 6: the first is "CCCCCC" which is a prefix of the concatenation of CCC and CCCI, the second is "MMMMMM" which is a prefix of the concatenation of MMM and MMMI;
a(30)=5, there is just one run with length 5: "XXXXX" which is contained in the concatenation of XIX, XX and XXI;
a(7)=a(644)=a(1359)=a(9375)=a(19194)=4, there are five runs with length 4: "IIII", two times "CCCC" and "MMMM", they occur in concatenations of (VIII, IX), (CC, CCI), (CCCXC, CCCXCI), (MM, MMI), (MMMCM, MMMCMI), respectively.

Reinhard Zumkeller, Table of n, a(n) for n = 1..19770
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A006968.

import Data.List (group)
a222581 n = a222581_list !! (n-1)
a222581_list = map length $ group a093796_list

A222581full = Map[Length, Split[Flatten[FromRomanNumeral[Characters[RomanNumeral[ Range[3999]]]]]]]; A222581full[[;;100]] (* Paolo Xausa, Mar 03 2024 *)

from itertools import groupby
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 r(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 [len(list(g)) for k, g in groupby("".join(r(i) for i in range(1, 4000)))]
print(afull()[:90]) # Michael S. Branicky, Mar 03 2024

A278182 Number of dots in Maya numeral representation of n.

Original entry on oeis.org

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

Offset: 0

José de Jesús Camacho Medina, Nov 20 2016

			a(1) = 1 because 1 in Maya numerals is "." (one dot).
a(2) = 2 because 2 in Maya numerals is ".." (two dots).

Wikipedia, Maya numerals

Cf. A006968.

Table[ Mod[n, 5] + Floor[Mod[n, 100]/20]  , {n, 0, 359}]

a(n) = (n mod 5) + floor(n mod 100 / 20) for 0<=n<=359.

cp's OEIS Frontend

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

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A093703 Numbers whose Roman numeral representation, reversed, is a Roman numeral.

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A131650 Number of symbols in Babylonian numeral representation of n.

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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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A105269 "Long" prime Roman numerals. Smallest prime whose Roman numeral representation has n characters.

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A105417 Numbers that are pandigital in Roman numerals, using each of the symbols I, V, X, L, C, D and M at least once.

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A116910 Numbers k such that the number of letters in the Roman numeral for k divides k.

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A142958 Roman numerals containing three letters (i.e., III, VII, XII, XIV, ...).

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