A006968 -id:A006968 - cp's OEIS frontend

A062726 Numbers that do not contain repeated letters when written in Roman numerals.

1, 4, 5, 6, 9, 10, 11, 14, 15, 16, 40, 41, 44, 45, 46, 50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 90, 91, 94, 95, 96, 100, 101, 104, 105, 106, 109, 110, 111, 114, 115, 116, 140, 141, 144, 145, 146, 150, 151, 154, 155, 156, 159, 160, 161, 164, 165, 166, 400, 401, 404

Offset: 1

Rodolfo Kurchan, Jul 11 2001

			9 is OK because when written in Roman numerals it is IX and has no letter repeated.
The largest possible term is a(316) = 1666 = MDCLXVI. - _Sean A. Irvine_, Apr 07 2023

Sean A. Irvine, Table of n, a(n) for n = 1..316
P. Lewis, Roman numerals and calendar

Cf. A006968.

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 23 2001

A082763 Roman numeral contains an asymmetric symbol (L).

Original entry on oeis.org

40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152

Offset: 1

Rick L. Shepherd, May 21 2003

I,V,X,C,D,M - and even the vinculum (bar) and apostrophus (backwards "C") - are each symmetric: horizontally, vertically, or both.

Numbers containing a 4 when decimally encoded with A061493. - Reinhard Zumkeller, Apr 14 2013

			40 = XL, 89 = LXXXIX, 140 = CXL.

Nathaniel Johnston, Table of n, a(n) for n = 1..2000 (complete up to 3999)
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numeral.

Cf. A006968 (Roman numerals main entry), A078715 (Palindromic Roman numerals).

a082763 n = a082763_list !! (n-1)
a082763_list = filter (containsL . a061493) [1..3999] where
   containsL x = d == 4 || x > 0 && containsL x' where
                 (x',d) = divMod x 10
-- Reinhard Zumkeller, Apr 14 2013

with(StringTools): for n from 1 to 152 do if(Search("L", convert(n, roman)) > 0)then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011

Select[Range[200],StringCases[RomanNumeral[#],"L"]!={}&] (* Harvey P. Dale, Jun 10 2023 *)

/* "%" use below is actually identical to lift(Mod(n-1,50)) */ /* (n-1)50 could be used for integer division below */ /* instead of floor, but the OEIS sometimes loses  */ /* characters depending upon where on a submitted line they are. */ a(n)=floor((n-1)/50)*100+40+(n-1)%50 for(n=1,125,print1(a(n),","))

a(n+50) = a(n) + 100 for n >= 1 [a(n+L) = a(n) + C for n >= I], a(1) = 40 [a(I) = XL], a(n+1) = a(n) + 1 for 1 <= n <= 49 [a(n+I) = a(n) + I for I <= n <= XLIX]; so a(n) = floor((n-1)/50)*100 + 40 + ((n-1)(mod 50)) for n >= 1 [a(n) = floor((n-I)/L)*C + XL + ((n-I)(mod L)) for n >= I].

A092197 Brevity advantage of "new style" over "old style" Roman numerals.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 2, 2, 2, 2, 4, 2, 2, 2, 2, 5, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 6, 0, 0, 0, 0, 2, 0

Offset: 1

Marc LeBrun, Feb 24 2004

			a(4)=2 because old style takes four letters (IIII) versus new style's two (IV).

Nathaniel Johnston, Table of n, a(n) for n = 1..3999

Cf. A006968, A092196.

A092197 := proc(n) return length(convert(n, roman, period=early)) - length(convert(n, roman)): end: seq(A092197(n),n=1..105); # Nathaniel Johnston, May 18 2011

a(n) = A092196(n) - A006968(n).

A093785 Numbers that are divisible by every digit in their Roman numeral representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 17 2004

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

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

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. A034838.

Cf. A061493.

a093785 n = a093785_list !! (n-1)
a093785_list = filter p [1..3999] where
   p v = q $ a061493 v where
     q w = w == 0 || v `mod` ([0,1,5,10,50,100,500,1000] !! d') == 0 && q w'
          where  (w',d) = divMod w 10; d' = fromInteger d
-- Reinhard Zumkeller, Apr 14 2013

A094109 Start with "I" in Roman numerals (one). The next number (in Roman numerals) describes the number of Roman numerals written previously in the sequence. Convert this infinite sequence into an infinite sequence of Arabic numbers.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 12, 15, 17, 21, 24, 28, 34, 39, 44, 48, 54, 57, 61, 64, 68, 74, 79, 84, 90, 92, 96, 100, 101, 103, 107, 111, 114, 118, 124, 129, 134, 140, 143, 149, 154, 158, 164, 169, 174, 180, 185, 191, 195, 199, 204, 208, 214, 219, 224, 230, 235, 241

Offset: 1

Eric Angelini, May 03 2004

			The sequence begins I, I, II, IV, VI, VIII, XII, XV, XVII, XXI, XXIV, XXVIII, XXXIV, XXIX, XLIII, XLVIII, LIV, LVII, LXI, LXIV, ...

Nathaniel Johnston, Table of n, a(n) for n = 1..573

See A006968 for how to spell the Roman numerals.

A094109 := proc(n) option remember: if(n<=2)then return 1:fi: return procname(n-1) + length(convert(procname(n-1),roman)): end: seq(A094109(n),n=1..58); # Nathaniel Johnston, May 18 2011

a(n) = a(n-1) + A006968(a(n-1)) for n >= 3. - Nathaniel Johnston, May 18 2011

Extended by Nathaniel Johnston, May 18 2011

A105247 Product of digits of Roman Numerals.

Original entry on oeis.org

1, 1, 1, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 50, 50, 50, 50, 50, 100, 100, 100, 100, 100, 500, 500, 500, 500, 500, 1000, 1000, 1000, 1000, 1000, 5000, 5000, 5000, 5000, 5000, 10000, 500, 500, 500, 500, 2500, 2500, 2500, 2500, 2500, 5000, 50, 50, 50, 50, 250

Offset: 1

Jonathan Vos Post, Apr 14 2005

New Roman Numerals A006968 (i.e., 4 = IV, not IIII). Related to sum of digits of n in Roman numeral representation A093783.

			a(3) = 1 because 3 = "III" and I*I*I = I = 1.
a(4) = 5 because 4 = "IV" and I*V = V = 5.
a(9) = 10 because 9 = "IX" and I*X = X = 10.
a(14) = 50 because 14 = "XIV" and X*I*V = L = 50.
a(19) = 100 because 19 = "XIX" and X*I*X = C = 100.
a(24) = 500 because 24 = "XXIV" and X*X*I*V = D = 500.
a(29) = 1000 because 29 = "XXIX" and X*X*I*X = M = 1000.
a(34) = 5000 because 34 = "XXXIV" and X*X*X*I*V = 5000.
a(39) = 10000 because 39 = "XXXIX" and X*X*X*I*X = myriad = 10000.
a(40) = 500 because 40 = "XL" and X*L = D = 500.
a(44) = 2500 because 44 = "XLIV" and X*L*I*V = MMD = 2500.
a(49) = 5000 because 49 = "XLIX" and X*L*I*X = 5000.
a(50) = 50 because 50 = L.

Nathaniel Johnston, Table of n, a(n) for n = 1..3999
Gerard Schildberger and Frank Ellermann, The numbers from 1 to 3999 expressed as Roman numerals.

Cf. A006968, A093783.

with(StringTools): A105247 := proc(n) local r: r:=convert(n, roman): return mul(parse(SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll( SubstituteAll(r[j], "I", "1"), "V", "5"), "X", "10"), "L", "50"), "C", "100"), "D", "500"), "M", "1000")),j=1..length(r)): end: seq(A105247(n),n=1..54); # Nathaniel Johnston, May 18 2011

A118121 Roman numeral complexity of n.

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 4, 3, 2, 3, 4, 5, 4, 3, 4, 5, 5, 4, 3, 4, 5, 5, 5, 4, 4, 5, 5, 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, 5, 6, 5, 4, 4, 5, 6, 5, 5, 6, 6, 6, 6, 2, 3, 4, 5, 4, 3, 4, 5, 6, 4, 1, 2, 3

Offset: 1

Jonathan Vos Post, May 12 2006

The least number of letters {I, V, X, L, C, D, M} needed to represent n by an expression with conventional Roman numerals, addition, multiplication and parentheses. a(n) <= A006968(n) and a(n) <= A005245(n). Conventional Roman numerals are very efficient at reducing complexity from number of letters in "old style" Roman numerals (A092196) and more primitive representations. In all but two examples shown (38, 88) the use of {+,*} reduces the representation by a single symbol (counting + and *); in these two it saves 2 symbols. In an alternate history, complexity theory and minimum description length could have been invented by Gregorius Catin.

			a(n) < A006968(n) for these examples. Here "<" means less in letter count:
a(18) = 4 [IX + IX < XVIII]; a(28) = 5 [XIV * II < XXVIII]; a(33) = 5 [XI * III < XXXIII]; a(36) = 4 [VI * VI < XXXVI]; a(37) = 5 [VI * VI + I < XXXVII]; a(38) = 5 [XIX * II < XXXVIII]; a(77) = 5 [XI * VII < LXXVII]; a(78) = 6 [XIII * VI < LXXVIII]; a(81) = 4 [IX * IX < LXXXI]; a(82) = 5 [XLI * II < LXXXII]; a(83) = 6 [XLI * II + I < LXXXIII]; a(84) = 5 [XX * IV < LXXXIV]; a(87) = 6 [IX * IX + VI < LXXXVII]; a(88) = 6 [XI * VIII < LXXXVIII].

Index to sequences related to the complexity of n

Cf. A005245, A006968, A092196.

A123060 Least positive integer k such that n has the same number of characters in base k and in Roman numeral representation, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 2, 2, 4, 11, 4, 3, 2, 3, 4, 3, 0, 2, 3, 5, 3, 0, 2, 0, 3, 0, 2, 0, 3, 4, 3, 0, 2, 0, 3, 0, 2, 0, 0, 7, 4, 3, 0, 3, 4, 3, 0, 2, 3, 51, 8, 4, 3, 4, 8, 4, 3, 0, 4, 8, 4, 3, 0, 3, 5, 3, 0, 0, 3, 5, 3, 0, 0, 0, 3, 0, 0, 2, 0, 3, 3, 0, 2, 0, 3, 0

Offset: 1

Jonathan Vos Post, Sep 26 2006

			a(1) = 1 since Roman(1) = I and 1(base 1) have the same (1) number of characters.
a(4) = 3 since Roman(4) = IV and 11(base 3) have the same (2) number of characters.
a(8) = 2 since Roman(8) = VIII and 1000(base 2) have the same (4) number of characters.
a(10) = 11 since Roman(10) = X and X(base 11) have the same (1) number of characters.
a(11) = 4 since Roman(11) = XI and 23(base 4) have the same (2) number of characters.
a(12) = 3 since Roman(12) = XII and 110(base 3) have the same (3) number of characters.
a(17) = 0 because Roman(17) = XVII has 4 characters, while 17 = 10001(base 2) has 5 characters and 17 = 122(base 3) has 3 characters.
a(30) = 4 because Roman(30) = XXX has 3 characters, as do 110(base 5) and 132(base 4), but Min{4,5} = 4.

Nathaniel Johnston, Table of n, a(n) for n = 1..3999
Gerard Schildberger, The first 3999 numbers in Roman numerals.

Cf. A006968.

A123060 := proc(n) local k,l,r: if(n<=3)then return 1:fi: r:=length(convert(n,roman)): for k from 2 to n+1 do l:=nops(convert(n,base,k)): if(l = r)then return k: elif(lA123060(n),n=1..86); # Nathaniel Johnston, May 18 2011

a(n) = min{k: StringLength(n base k) = StringLength(Roman(n))}, or 0 if no such k exists. a(n) = min{k: A006968(n) = 1 + floor(log_b(n))}, or 0 if no such k exists.

Extended and corrected by Nathaniel Johnston, May 18 2011

A130228 Roman numerals with "i" replaced by "1", "v" replaced by "5", "x" replaced by 10, etc., sorted in increasing order.

Original entry on oeis.org

1, 5, 10, 11, 15, 50, 51, 100, 101, 105, 110, 111, 500, 501, 505, 511, 1000, 1001, 1005, 1010, 1011, 1015, 1050, 1051, 5001, 5005, 5010, 5011, 5015, 5051, 5111, 10001, 10005, 10010, 10011, 10015, 10050, 10051, 10100, 10101, 10105, 10110, 10111

Offset: 1

Eric Angelini, Aug 05 2007

Cf. A093788.

Eric Angelini, Table of n, a(n) for n = 1..153
Gerard Schildberger, The first 3999 numbers in Roman numerals

A242181 Numbers with four X's in Roman numerals.

Original entry on oeis.org

39, 89, 139, 189, 239, 289, 339, 389, 439, 489, 539, 589, 639, 689, 739, 789, 839, 889, 939, 989, 1039, 1089, 1139, 1189, 1239, 1289, 1339, 1389, 1439, 1489, 1539, 1589, 1639, 1689, 1739, 1789, 1839, 1889, 1939, 1989, 2039, 2089, 2139, 2189, 2239, 2289, 2339

Offset: 1

J. Lowell, May 06 2014

All these end in XXXIX.

			39 = XXXIX; 89 = LXXXIX.

Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006968.

A242181:=n->50*n-11; seq(A242181(n), n=1..50); # Wesley Ivan Hurt, May 07 2014
[seq(convert(50*n-11, roman), n=1..50)]; # for the Roman version - Wolfdieter Lang, May 09 2014

Table[50 n - 11, {n, 50}] (* Wesley Ivan Hurt, May 07 2014 *)

a(n)=50*n-11 \\ Charles R Greathouse IV, May 06 2014

a(n) = 50n - 11. - Charles R Greathouse IV, May 06 2014

More terms from Wesley Ivan Hurt, May 07 2014

cp's OEIS Frontend

A062726 Numbers that do not contain repeated letters when written in Roman numerals.

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A082763 Roman numeral contains an asymmetric symbol (L).

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A092197 Brevity advantage of "new style" over "old style" Roman numerals.

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A093785 Numbers that are divisible by every digit in their Roman numeral representation.

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A094109 Start with "I" in Roman numerals (one). The next number (in Roman numerals) describes the number of Roman numerals written previously in the sequence. Convert this infinite sequence into an infinite sequence of Arabic numbers.

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Maple

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A105247 Product of digits of Roman Numerals.

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Maple

A118121 Roman numeral complexity of n.

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A123060 Least positive integer k such that n has the same number of characters in base k and in Roman numeral representation, or 0 if no such k exists.

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