A138166 -id:A138166 - cp's OEIS frontend

A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063

Offset: 1

Patrick De Geest, Nov 15 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A052019, A052020, A052021, A052022, A007953, A005349, A028834.
Cf. A175688. - Reinhard Zumkeller, Aug 15 2010
Cf. A055642, A004426.
Cf. A119246, A138166.

import Data.List (isInfixOf)
a052018 n = a052018_list !! (n-1)
a052018_list = filter f [0..] where
   f x = show (a007953 x) `isInfixOf` show x
-- Reinhard Zumkeller, Jun 18 2013

sdssQ[n_]:=Module[{idn=IntegerDigits[n],s,len},s=Total[idn];len= IntegerLength[ s]; MemberQ[Partition[idn,len,1],IntegerDigits[s]]]; Join[{0},Select[Range[1100],sdssQ]] (* Harvey P. Dale, Jan 02 2013 *)

loop = (str(n) for n in range(399))
print([int(n) for n in loop if str(sum(int(k) for k in n)) in n]) # Jonathan Frech, Jun 05 2017

A119246 Numbers containing in decimal representation their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 29, 30, 39, 40, 49, 50, 59, 60, 69, 70, 79, 80, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 109, 118, 127, 128, 136, 138, 145, 148, 154, 158, 163, 168, 172, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198, 199, 200

Offset: 1

Reinhard Zumkeller, May 10 2006

Complement of A119247.

For terms u: all digital permutations of u form terms; u*10 and all insertions of 0 are terms; if v is another term, then the concatenations uv, vu are also terms, as well as all insertions of v in u; these properties allow the construction of all terms beginning with {d:1<=d<=9}. - Reinhard Zumkeller, May 19 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

Cf. A052018, A031298, A138166.

a119246 n = a119246_list !! (n-1)
a119246_list =
    filter (\x -> a010888 x `elem` a031298_row (fromInteger x)) [0..]
-- Reinhard Zumkeller, Dec 16 2013, Apr 14 2011

d[n_] := IntegerDigits[n]; Select[Range[0, 200], MemberQ[d[#1], NestWhile[Total[d[#]] &, #1, # > 9 &]] &] (* Jayanta Basu, Jul 13 2013 *)

A038528 If n has decimal expansion abc...d, with k digits, let f(n) be obtained by deleting all k's from abc...d, closing up and deleting initial 0's; sequence gives n such that f(f(f(...(n)))) = 0 or empty.

Original entry on oeis.org

1, 12, 20, 21, 22, 123, 132, 133, 203, 213, 223, 230, 231, 232, 300, 301, 303, 312, 313, 320, 321, 322, 330, 331, 333, 1234, 1243, 1244, 1324, 1334, 1342, 1343, 1423, 1424, 1432, 1433, 1442, 1444, 2034, 2043, 2044, 2134, 2143, 2144, 2234

Offset: 1

Naohiro Nomoto

The sequence has exactly 14174521 terms, 999999999 is the last and largest. - Reinhard Zumkeller, Jul 04 2012

			If n=22 (2 digits), f(n) = empty. If n=230 (3 digits), f(n)=20, f(f(n))=0. If n=301 (3 digits), f(n)=1 (1 digit), f(f(n))=empty.
The last 12 terms are: 999999333, 999999900, 999999901, 999999909, 999999912, 999999919, 999999920, 999999921, 999999922, 999999990, 999999991, 999999999.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A038527.

Cf. A002024, A055642, A031298, subsequence of A138166.

import Data.List ((\\))
a038528 n = a038528_list !! (n-1)
a038528_list = gen ([1], 1) where
   gen (_, 10) = []
   gen (ds, len)
      | len `elem` ds && chi ds
        = foldr (\u v -> u + 10*v) 0 ds : gen (succ (ds, len))
      | otherwise = gen (succ (ds, len))
   chi xs = null ys || ys /= xs && chi ys where
            ys = tr $ filter (/= length xs) xs
            tr zs = if null zs || last zs > 0 then zs else tr $ init zs
   succ ([], len)   = ([1], len + 1)
   succ (d : ds, len)
       | d < len = (head (dropWhile (<= d) a002024_list \\ ds) : ds, len)
       | otherwise = (0 : ds', len') where (ds', len') = succ (ds, len)
-- Reinhard Zumkeller, Jul 04 2012

zeroQ[n_] :=  FixedPoint[ Function[{k}, DeleteCases[id = IntegerDigits[k], Length[id]] // FromDigits[#, 10]&], n] == 0; Select[Range[10^4], zeroQ] (* Jean-François Alcover, Dec 10 2014 *)

A054055(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Jul 04 2012

A138167 Numbers containing their length in ternary representation.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 10, 11, 12, 21, 31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 49, 58, 66, 67, 68, 76, 86, 95, 96, 97, 98, 104, 113, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

Reinhard Zumkeller, Mar 03 2008

			42 ->'1120', length = 4 ->'11', therefore 42 is a term;
420 ->'120120', length = 6 ->'20', therefore 420 is a term.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081604, A138166, A138168.

A138168 Numbers containing their length in binary representation.

Original entry on oeis.org

1, 2, 6, 7, 8, 9, 12, 20, 21, 22, 23, 26, 27, 29, 38, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 71, 78, 79, 87, 92, 93, 94, 95, 103, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

Offset: 1

Reinhard Zumkeller, Mar 03 2008

			20 ->'10100', length = 5 ->'101', therefore 20 is a term;
200 ->'11001000', length = 8 ->'1000', therefore 200 is a term.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A070939, A138166, A138167.

lbrQ[n_]:=Module[{idn2=IntegerDigits[n,2]},SequenceCount[idn2, IntegerDigits[ Length[ idn2],2]]>0]; Select[Range[200],lbrQ] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Oct 07 2015 *)

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A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

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A119246 Numbers containing in decimal representation their digital root.

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A038528 If n has decimal expansion abc...d, with k digits, let f(n) be obtained by deleting all k's from abc...d, closing up and deleting initial 0's; sequence gives n such that f(f(f(...(n)))) = 0 or empty.

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A138167 Numbers containing their length in ternary representation.

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Keywords

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A138168 Numbers containing their length in binary representation.

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Author

Keywords

Examples

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Mathematica