A073327 -id:A073327 - cp's OEIS frontend

A075831 Like A073327, but multiply the numerical values of the letters instead of adding them.

35100, 1050, 6900, 72000, 34020, 5940, 4104, 146300, 50400, 8820, 1400, 462000, 3036000, 181440000, 238140000, 2268000, 28728000, 1024100000, 17640000, 61740000, 16100000, 16905000000, 111090000000, 1159200000000, 547722000000

Offset: 0

Derek McHugh (derek.mchugh(AT)transwareplc.com), Oct 14 2002

			Here a=1, b=2, c=3, d=4, etc., a(17) = s*e*v*e*n*t*e*e*n = 19*5*22*5*14*20*5*5*14 = 1024100000.
"One" -> 15*14*5 = 1050.

More terms from Matthew Goers, Oct 26 2009, Nov 03 2009
Edited by N. J. A. Sloane, Nov 12 2009
Offset corrected by Sean A. Irvine, Mar 10 2025

A152611 Assign weights to the nonnegative integers as in A073327, then sort them by weight.

Original entry on oeis.org

1, 10, 5, 9, 8, 6, 3, 2, 4, 11, 0, 7, 15, 50, 18, 80, 40, 19, 12, 90, 16, 60, 13, 30, 51, 14, 20, 1000000000, 55, 59, 81, 100, 17, 70, 58, 5000000000, 9000000000, 85, 89, 500, 900, 1000000000000000000000000000000000, 41, 56, 91, 53, 8000000000, 88, 800, 52

Offset: 1

Matthew Goers, Oct 26 2009, Nov 02 2009, Nov 03 2009

We are using U.S. spelling, as in A108067 rather than A004740, and we ignore hyphens and spaces.

In the case of ties, sort by numerical value.

			Let wt(n) = A073327(n) denote the weight of n. We have wt(1) = 34 < wt(10) = 39 < wt(5) = 42 = wt(9) = 42 < wt(8) = 49 < ...

A345126 a(1) = 1; thereafter, a(n) = A073327(a(n-1)), using the British English version of A073327 when there is a choice.

Original entry on oeis.org

1, 34, 160, 224, 318, 222, 316, 245, 277, 326, 308, 198, 263, 304, 209, 193, 270, 261, 282, 283, 281, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259, 259

Offset: 1

Paul Duckett, Jun 08 2021

This sequence uses UK English names (e.g., one hundred and one).

Theorem: All nonnegative integers eventually reach either the fixed point 251, the fixed point 259, or the five-cycle 248, 284, 285, 267, 313.

			a(1) =   1 = ONE                   => A073327(1)   =  34,
a(2) =  34 = THIRTY FOUR           => A073327(34)  = 160,
a(3) = 160 = ONE HUNDRED AND SIXTY => A073327(160) = 224, and so on.

Cf. A073327, A345157.

A345157 a(1) = 1; thereafter, a(n) = A073327(a(n-1)), using the US English version of A073327 when there is a choice.

Original entry on oeis.org

1, 34, 160, 205, 174, 278, 291, 253, 254, 258, 247, 281, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240, 216, 228, 288, 255, 240

Offset: 1

Paul Duckett, Jun 09 2021

This is a US English version of A345126 (i.e., not using AND).

The last five integers cycle. This is the only cycle [proved], so all nonnegative integers chain to enter it.

			1 = ONE = 34, 34 = THIRTY FOUR = 160, 160 = ONE HUNDRED SIXTY = 205.

Cf. A073327, A345126.

A072959 Using the US English names for the nonnegative integers, assign each letter a numerical value as in A073327 (A=1, B=2, ..., Z=26), treat the name as a base-27 integer, and convert to decimal.

Original entry on oeis.org

515904, 11318, 15216, 10799546, 129618, 125258, 14118, 10211981, 2839691, 282506, 14729, 78236429, 299309045, 212445531527, 68884716992, 2457249197, 7503281492, 5427065792075, 55893641747, 150135668600, 299310469

Offset: 0

Michael Joseph Halm, Aug 13 2002

For names with spaces (e.g., ONE HUNDRED), treat each space as a '0', or placeholder, in the base-27 system. (Therefore ONE HUNDRED = 3196540902115084.)

English name for the number n transliterated into Lee Sallows's base-27 system.

			a(1) = 11318 because o(729) + n(27) + e = 10935 + 378 + 5 = 11318.
a(2) = 15216 because "TWO" in base 27 gives 20*27^2 + 23*27 + 15 = 15216.

M. J. Halm, Sequences (Re)discovered, Mpossibilities 81 (Aug. 2002).

M. J. Halm, Jootsy Calculus.

lSallow27 := proc(s)
local a,i,c ;
a := 0 ;
for i from 1 to length(s) do
c := substring(s,i) ;
if c = " " then
a := 27*a ;
else
a := 27*a + StringTools[Ord](c) -96 ;
fi;
od:
a ;
end:
enums := ["one","two","three","four","five","six","seven","eight","nine","ten",
"eleven","twelve", "thirteen","fourteen","fifteen","sixteen","seventeen",
"eighteen","nineteen","twenty"]:
for i from 1 to nops(enums) do
printf("%d %d\n",i, lSallow27(enums[i])) ;
od:
# R. J. Mathar

In Sallows's system, space = 0, A = 1, B = 2, etc. to Z = 26, so that words and phrases, even number names, can be transformed into numbers.

Definition rephrased by Matthew Goers, Nov 03 2009
The old version of this sequence was wrong. Don Reble and R. J. Mathar supplied a corrected version. Edited by N. J. A. Sloane, Sep 20 2009
Edited by N. J. A. Sloane, Aug 15 2010 at the suggestion of D. S. McNeil
Offset corrected by Sean A. Irvine, Nov 07 2024

A226911 Remainder modulo n of the sum of the letters of the English word(s) for n (A073327: a=1, ..., z=26).

Original entry on oeis.org

0, 0, 2, 0, 2, 4, 2, 1, 6, 9, 8, 3, 8, 6, 5, 0, 7, 1, 10, 7, 15, 11, 2, 23, 24, 3, 10, 16, 4, 10, 10, 30, 24, 24, 2, 8, 17, 35, 25, 4, 36, 16, 11, 12, 36, 44, 8, 37, 28, 16, 49, 20, 16, 18, 53, 6, 17, 57, 49, 37, 9, 31, 27, 29, 9, 17, 28, 10, 1, 40, 2, 24, 20, 22, 2, 10, 21, 3

Offset: 1

Eric Angelini and M. F. Hasler, Jun 22 2013

By definition, a(n) < n so iterated application of this function to any initial value n will create a strictly decreasing sequence ending in 0.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Hasler in reply to E. Angelini, English number words modulo themselves, SeqFan list, Jun 21 2013

Cf. A073029, A119945, A072922, A075831, A152611, A052360, A362065.

f:= proc(n) local S;
   uses StringTools;
  S:= Select(IsAlpha,convert(n,english));
  convert(map(`-`,convert(S,bytes),96),`+`) mod n
end proc:
map(f, [$1..100]); # Robert Israel, Jun 12 2019

a[n_] := Mod[Total@ Flatten[ ToCharacterCode[#] - 96 & /@ Characters@ StringDelete[ IntegerName[n], Except@ LetterCharacter]], n] (* after Michael De Vlieger in A362065 *); Array[a, 78] (* Robert G. Wilson v, Apr 22 2023 *)

PARI
```
A226911 = n->A073327(n)%n
```

a(n) = A073327(n) mod n.

It appears that a(n) = A073327(n) for n > 279. - Robert Israel, Jun 12 2019

A004740 Integers in alphabetical order in U.S. English.

Original entry on oeis.org

8, 8000000000, 8000000008, 8000000018, 8000000080, 8000000088, 8000000085

Offset: 1

N. J. A. Sloane, Neil Fernandez

Jasper Mulder, Jan 25 2010, comments that this sequence is ill-defined, since there are an infinite number of cardinal numbers that start with a 'd', namely all those powers of 10 named decillion, duodecillion and so on (see http://en.wikipedia.org/wiki/English_numerals).

However, it would seem then that 'billion' and 'centillion' also precede these, while it is standard (and systematic for this sequence) to refer to such numbers as 'one billion', 'one decillion', etc. - Yasiru Ratnayake, May 03 2012

			The list begins "eight", "eight billion", "eight billion eight", ... [corrected by _Paul Duckett_, Nov 26 2023]

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Cf. A019440, A026081. See A108067 for another version.

Cf. A073327, A152611, A075831, A000052.

A169639 Write n in French (cf. A167507) and sum the letters using a=1, ..., z=26.

Original entry on oeis.org

64, 35, 54, 81, 82, 43, 52, 60, 58, 46, 37, 60, 71, 83, 123, 92, 64, 97, 95, 83, 72, 132, 126, 153, 154, 115, 124, 132, 130, 118, 82, 142, 136, 163, 164, 125, 134, 142, 140, 128, 97, 157, 151, 178, 179, 140, 149, 157, 155, 143, 104, 164, 158, 185, 186, 147, 156, 164

Offset: 0

Jacques ALARDET, Apr 04 2010

Accents are ignored. - Chai Wah Wu, Jun 11 2021

			From _Omar E. Pol_, Jun 15 2021: (Start)
-----------------------------------------------------
   n      Name      Calculation                  a(n)
-----------------------------------------------------
   0      Zero      26 +  5 + 18 + 15           = 64
   1      Un        21 + 14                     = 35
   2      Deux       4 +  5 + 21 + 24           = 54
   3      Trois     20 + 18 + 15 +  9 + 19      = 81
   4      Quatre    17 + 21 +  1 + 20 + 18 +  5 = 82
   5      Cinq       3 +  9 + 14 + 17           = 43
   6      Six       19 +  9 + 24                = 52
   7      Sept      19 +  5 + 16 + 20           = 60
   8      Huit       8 + 21 +  9 + 20           = 58
   9      Neuf      14 +  5 + 21 +  6           = 46
  10      Dix        4 +  9 + 24                = 37
  11      Onze      15 + 14 + 26 +  5           = 60
  12      Douze      4 + 15 + 21 + 26 +  5      = 71
... (End)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A167507, A169641 ("limit points").

For other languages see A073327 (U.S. English), A119945 (German), A161406 (Spanish).

A169639(n)=vecsum([t%32|t<-Vecsmall(French(n)),t>64]) \\ Cf. A167507 for French(). - M. F. Hasler, Apr 08 2023

from num2words import num2words
from unidecode import unidecode
def A169639(n): return sum(ord(s)-96 for s in unidecode(num2words(n,lang='fr')) if s.isalpha()) # Chai Wah Wu, Jun 11 2021

Edited by N. J. A. Sloane, Apr 05 2010

Offset corrected by Mohammed Yaseen, Jun 11 2021

A073029 Names for numbers in American English, with each letter transformed into its index in the alphabet.

Original entry on oeis.org

26, 5, 18, 15, 15, 14, 5, 20, 23, 15, 20, 8, 18, 5, 5, 6, 15, 21, 18, 6, 9, 22, 5, 19, 9, 24, 19, 5, 22, 5, 14, 5, 9, 7, 8, 20, 14, 9, 14, 5, 20, 5, 14, 5, 12, 5, 22, 5, 14, 20, 23, 5, 12, 22, 5, 20, 8, 9, 18, 20, 5, 5, 14, 6, 15, 21, 18, 20, 5, 5, 14, 6, 9, 6, 20, 5, 5, 14, 19, 9, 24, 20, 5

Offset: 0

Zak Seidov, Aug 22 2002

Irregular triangle read by rows, in which row n lists the successive indices of the letters in the American English name for n. For example, row one is 15, 14, 5. - N. J. A. Sloane, Apr 22 2023
Find smallest n's for which a(n)=1,2,3,4,...,26.
A: The numbers 10 and 11 never occur. The rows in which the others occur first (assuming use of the "short scale") are 1000 (thousAnd), 10^9 (Billion), 10^27 (oCtillion), 100 (hunDred), 0 (zEro), 4 (Four), 8 (eiGht), 3 (tHree), 5 (fIve), --, -- (j & k don't occur in English names of numbers), 11 (eLeven), 10^6 (Million), 1 (oNe), 0 (zerO), 10^24 (sePtillion), 10^15 (Quadrillion), 0 (zeRo), 6 (Six), 2 (Two), 4 (foUr), 5 (fiVe), 2 (tWo), 6 (siX), 20 (twentY), 0 (Zero). Converting the position in the row plus the preceding row lengths to a linear index n this yields (after subtracting 1 to match offset 0 of the sequence): 18452, ?, ?, 864, 1, 15, 33, 11, 20, -, -, 44, ?, 5, 3, ?, ?, 2, 23, 7, 17, 21, 8, 25, 115, 0. The graph nicely shows the position & frequency of the individual letters. - M. F. Hasler, Feb 06 2016

			Top row is for "zero"; "z" is the 26th letter in the alphabet, "e" the fifth, "r" the 18th and "o" the 15th, so the first row is 26,5,18,15.

M. F. Hasler, Table of n, a(n) for n = 0..18454 (rows 0..1000 "flattened"), Feb 06 2016
M. F. Hasler, rows 0 - 1000, Feb 06 2016
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens, Jun 08 2010

Cf. A005589 (row lengths).
Cf. A073327 (row sums).
Cf. A072922.

alef=Characters["abcdefghijklmnopqrstuvwxyz"]; numb="zeroonetwothreefourfivesixseveneightnineteneleven[...]"; ch=Characters[numb]; seq[n_] := Position[alef, ch[[n]]][[1, 1]] (* Franklin T. Adams-Watters, Jun 02 2006 *)
(* see the link to a001477.txt, copy the first 17 lines and then paste and assign to the variable 'str' as a string as follows *)
str = "zeroonetwothreefourfivesixseveneightnineteneleventwelvethirteenfourteenfifteensixteen";
Characters@ str /. Flatten[ Table[ {FromCharacterCode[96 + n] -> n}, {n, 26}]] (* Robert G. Wilson v, Jun 08 2010 *)

A073029_row(n)=select(t->t>64,Vec(Vecsmall(English(n))))%32 \\ See A052360 for English(). - M. F. Hasler, Feb 06 2016

from num2words import num2words
def row(n): return [ord(c)-96 for c in num2words(n).replace(" and", "") if c.isalpha()]
print([e for n in range(17) for e in row(n)]) # Michael S. Branicky, Apr 22 2023

Corrected and extended by Franklin T. Adams-Watters, Jun 02 2006, Oct 24 2006

A119945 Sum of numerical values of letters in German names of the nonnegative numbers.

Original entry on oeis.org

59, 47, 63, 36, 54, 52, 54, 54, 32, 54, 53, 23, 87, 89, 107, 105, 88, 88, 85, 107, 106, 173, 208, 181, 199, 197, 199, 199, 177, 199, 90, 157, 192, 165, 183, 181, 183, 183, 161, 183, 96, 163, 198, 171, 189, 187, 189, 189, 167, 189, 94, 161, 196, 169, 187, 185, 187, 187, 165

Offset: 0

Wolfdieter Lang, Jul 20 2006

Letters with umlauts are interpreted respectively as "ae" (which does not appear in this entry), "oe", "ue", and "sz" (sharp s) (as in the German name of 30 = "dreißig" -> "dreissig") as "ss" (not "sz").
According to the Reder reference only "zweihundertfuenf", 205 and "zweihundertsieben", 207, satisfy a(n)=n.
This sequence is ambiguous for numbers above 100 because one can use, for instance, for 101 "hundertundeins" or "hunderteins. To avoid such ambiguities one should always stick to the shorter version.
An alternate version of this sequence could ignore umlauts (i.e., take "a" for "ä" etc), or, more in-line with the German alphabet as it is usually listed in textbooks and reference works, taken as ä=27, ö=28, ü=29 (and then maybe ß=30, which could nonetheless remain considered as a ligature of "∫s"="ss"). - M. F. Hasler, Jun 23 2013
It appears that there is no canonical version of this sequence, because of the lack of agreement even on the number of letters in the German alphabet. - N. J. A. Sloane, Jun 11 2021

			"Null" for 0 in German has numerical values (a=1, b=2, ..., z=26) [14, 21, 12, 12] which sums up to a(0)=59.
The numerical values for "zweihundertfuenf" are [26, 23, 5, 9, 8, 21, 14, 4, 5, 18, 20, 6, 21, 5, 14, 6] with sum 205.
The numerical values for "zweihundertsieben" are [26, 23, 5, 9, 8, 21, 14, 4, 5, 18, 20, 19, 9, 5, 2, 5, 14] with sum 207.
From _Omar E. Pol_, Jun 15 2021: (Start)
-------------------------------------------------------------
   n     Name               Calculation                  a(n)
-------------------------------------------------------------
   0     Null               14 + 21 + 12 + 12           = 59
   1     Eins                5 +  9 + 14 + 19           = 47
   2     Zwei               26 + 23 +  5 +  9           = 63
   3     Drei                4 + 18 +  5 +  9           = 36
   4     Vier               22 +  9 +  5 + 18           = 54
   5     Fünf  --> Fuenf     6 + 21 +  5 + 14 +  6      = 52
   6     Sechs              19 +  5 +  3 +  8 + 19      = 54
   7     Sieben             19 +  9 +  5 +  2 +  5 + 14 = 54
   8     Acht                1 +  3 +  8 + 20           = 32
   9     Neun               14 +  5 + 21 + 14           = 54
  10     Zehn               26 +  5 +  8 + 14           = 53
  11     Elf                 5 + 12 +  6                = 23
  12     Zwölf --> Zwoelf   26 + 23 + 15 +  5 + 12 +  6 = 87
... (End)
a(16) = 88 because "sechzehn" => [19, 5, 3, 8, 26, 5, 8, 14] with sum 88, as for a(17) with "siebzehn" => [19, 9, 5, 2, 26, 5, 8, 14]. - _M. F. Hasler_, Apr 08 2023

Christian Reder, Wörter und Zahlen, Springer Verlag, Komet, Wien, 2000, p. 337.

Index entries for sequences related to German names of numbers

For analogs in other languages see A073327 (U.S. English), A169639 (French), A161406 (Spanish).

G(n, eins="eins")={my(s(n, p, z, e="ein")=n=divrem(n, p); if(n[2], Str(G(n[1]*p), G(n[2])), Str(G(n[1], e), z))); if(n<20, ["null", eins, "zwei", "drei", "vier", "fuenf", "sechs", "sieben", "acht", "neun", "zehn", "elf", "zwoelf", "dreizehn", "vierzehn", "fuenfzehn", "sechzehn", "siebzehn", "achtzehn", "neunzehn"][n+1], n<100 && n%10, Str(G(n%10, "ein"), "und", G(n\10*10)), n<100, ["zwanzig", "dreissig", "vierzig", "fuenfzig", "sechzig", "siebzig", "achtzig", "neunzig"][n\10-1], n<1000, s(n, 100, "hundert"), n<10^6, s(n, 1000, "tausend"), n<10^9, s(n, 10^6, if(n\10^6>1, " Millionen ", "e Million ")), n<10^12, s(n, 10^9, if(n\10^9>1, " Milliarden ", "e Milliarde ")))}
\\ extension to Billion, Billiarde, Trillion, Trilliarde, ... is obvious. See A007208 for a variant.
apply( {A119945(n)=vecsum([t%32|t<-Vecsmall(G(n)),t>64])}, [0..99])
\\ M. F. Hasler, Apr 08 2023

Edited by N. J. A. Sloane, Jun 10 2021

Corrected and extended by M. F. Hasler, Apr 08 2023

cp's OEIS Frontend

A075831 Like A073327, but multiply the numerical values of the letters instead of adding them.

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A152611 Assign weights to the nonnegative integers as in A073327, then sort them by weight.

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Examples

A345126 a(1) = 1; thereafter, a(n) = A073327(a(n-1)), using the British English version of A073327 when there is a choice.

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A345157 a(1) = 1; thereafter, a(n) = A073327(a(n-1)), using the US English version of A073327 when there is a choice.

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A072959 Using the US English names for the nonnegative integers, assign each letter a numerical value as in A073327 (A=1, B=2, ..., Z=26), treat the name as a base-27 integer, and convert to decimal.

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A226911 Remainder modulo n of the sum of the letters of the English word(s) for n (A073327: a=1, ..., z=26).

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A004740 Integers in alphabetical order in U.S. English.

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A169639 Write n in French (cf. A167507) and sum the letters using a=1, ..., z=26.

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A073029 Names for numbers in American English, with each letter transformed into its index in the alphabet.

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