A031298 -id:A031298 - cp's OEIS frontend

A261793 Successively add the smallest number that is not a substring in decimal representation.

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Reinhard Zumkeller, Sep 01 2015

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

Cf. A031298, A261794, A261795 (first differences), A261806, A260273, A261786

a261793 n = a261793_list !! (n-1)
a261793_list = iterate (\x -> x + a261794 x) 1

A080719 Replace decimal digits with their binary values and convert back to decimal representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 6, 7, 12, 13, 14, 15, 24, 25, 4, 5, 10, 11, 20, 21, 22, 23, 40, 41, 6, 7, 14, 15, 28, 29, 30, 31, 56, 57, 8, 9, 18, 19, 36, 37, 38, 39, 72, 73, 10, 11, 22, 23, 44, 45, 46, 47, 88, 89, 12, 13, 26, 27, 52, 53, 54, 55, 104, 105, 14, 15, 30, 31, 60, 61, 62

Offset: 0

Reinhard Zumkeller, Mar 06 2003

m is a local maximum iff m == 9 modulo 10 (see A017377).

A257831 seen as binary numbers: A007088(a(n)) = A257831(n). - Reinhard Zumkeller, May 10 2015

			n=27 -> '2''7' -> '10''111' -> '10111' -> 23: a(27)=23.
See also A257831.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088.

Cf. A257831, A030308, A031298.

import Data.Maybe (mapMaybe)
a080719 = foldr (\b v -> 2 * v + b) 0 .
           concat . mapMaybe (flip lookup bin) . a031298_row
            where bin = zip [0..9] a030308_tabf
-- Reinhard Zumkeller, May 10 2015

Table[FromDigits[Flatten[IntegerDigits[#,2]&/@IntegerDigits[n]],2],{n,80}] (* Harvey P. Dale, Aug 30 2014 *)

def A080719(n):
    return int(''.join((format(int(d),'b') for d in str(n))),2)
# Chai Wah Wu, May 10 2015

a(0)=0 prepended and offset changed by Reinhard Zumkeller, May 10 2015

A261794 a(n) is the smallest nonzero number that is not a substring of n in decimal representation.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Sep 01 2015

A261795(n) = a(A261793(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007089, A031298, A261795, A261793, A261461, A261787.

import Data.List (isInfixOf)
a261794 x = f $ tail a031298_tabf where
   f (cs:css) = if isInfixOf cs (a031298_row x)
                   then f css else foldr (\d v -> 10 * v + d) 0 cs

Name corrected by Álvar Ibeas, Sep 08 2020

A262437 Triangle T(n,k): write n in base 16, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 15, 2

Offset: 0

Reinhard Zumkeller, Sep 22 2015

Sum(T(n,k)*16^k : k = 0..A262438(n)-1) = n.

			.   n | HEX | T(n,*)        n | HEX | T(n,*)        n | HEX | T(n,*)
. ----+-----+--------     ----+-----+--------     ----+-----+--------
.   0 |   0 |  [0]         24 |  18 |  [8,1]       48 |  30 |  [0,3]
.   1 |   1 |  [1]         25 |  19 |  [9,1]       49 |  31 |  [1,3]
.   2 |   2 |  [2]         26 |  1A |  [10,1]      50 |  32 |  [2,3]
.   3 |   3 |  [3]         27 |  1B |  [11,1]      51 |  33 |  [3,3]
.   4 |   4 |  [4]         28 |  1C |  [12,1]      52 |  34 |  [4,3]
.   5 |   5 |  [5]         29 |  1D |  [13,1]      53 |  35 |  [5,3]
.   6 |   6 |  [6]         30 |  1E |  [14,1]      54 |  36 |  [6,3]
.   7 |   7 |  [7]         31 |  1F |  [15,1]      55 |  37 |  [7,3]
.   8 |   8 |  [8]         32 |  20 |  [0,2]       56 |  38 |  [8,3]
.   9 |   9 |  [9]         33 |  21 |  [1,2]       57 |  39 |  [9,3]
.  10 |   A |  [10]        34 |  22 |  [2,2]       58 |  3A |  [10,3]
.  11 |   B |  [11]        35 |  23 |  [3,2]       59 |  3B |  [11,3]
.  12 |   C |  [12]        36 |  24 |  [4,2]       60 |  3C |  [12,3]
.  13 |   D |  [13]        37 |  25 |  [5,2]       61 |  3D |  [13,3]
.  14 |   E |  [14]        38 |  26 |  [6,2]       62 |  3E |  [14,3]
.  15 |   F |  [15]        39 |  27 |  [7,2]       63 |  3F |  [15,3]
.  16 |  10 |  [0,1]       40 |  28 |  [8,2]       64 |  40 |  [0,4]
.  17 |  11 |  [1,1]       41 |  29 |  [9,2]       65 |  41 |  [1,4]
.  18 |  12 |  [2,1]       42 |  2A |  [10,2]      66 |  42 |  [2,4]
.  19 |  13 |  [3,1]       43 |  2B |  [11,2]      67 |  43 |  [3,4]
.  20 |  14 |  [4,1]       44 |  2C |  [12,2]      68 |  44 |  [4,4]
.  21 |  15 |  [5,1]       45 |  2D |  [13,2]      69 |  45 |  [5,4]
.  22 |  16 |  [6,1]       46 |  2E |  [14,2]      70 |  46 |  [6,4]
.  23 |  17 |  [7,1]       47 |  2F |  [15,2]      71 |  47 |  [7,4]
.  24 |  18 |  [8,1]       48 |  30 |  [0,3]       72 |  48 |  [8,4]  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Hexadecimal
Wikipedia, Hexadecimal

Cf. A001025, A262438 (row lengths), A030308 (binary), A030341 (ternary), A031298 (decimal).

a262437 n k = a262437_tabf !! n !! k
a262437_row n = a262437_tabf !! n
a262437_tabf = iterate succ [0] where
   succ []      = [1]
   succ (15:hs) = 0 : succ hs
   succ (h:hs)  = (h + 1) : hs

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

A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554

Offset: 1

Amarnath Murthy, Aug 10 2005

Except for initial 0, rearrangement of numbers in A056524. They first differ at a(101) = 110011, while A056524(101) = 101101. If n has digits d_1 d_2 ... d_k, permute them to d_1 d_k d_2 d_{k-1} ... d_{floor(k/2)+1} and use that as index to A056524. - Franklin T. Adams-Watters, Jun 20 2006

			a(12) = 1221, deleting the LSB and the third digit 2 we get 12, deleting second and fourth digit we get 21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A045918.

Cf. A056524, A031298.

import Data.List (transpose)
a110745 n = read (concat $ transpose [ns, reverse ns]) :: Integer
            where ns = show n
-- Reinhard Zumkeller, Feb 14 2015

More terms from Franklin T. Adams-Watters, Jun 20 2006

A249626 a(0) = 0, a(n+1) = smallest number, not occurring earlier, containing the smallest of the least frequently occurring digits in all preceding terms.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 40, 45, 46, 47, 48, 49, 50, 56, 57, 58, 59, 60, 67, 68, 69, 70, 78, 79, 80, 89, 90, 100, 21, 31, 41, 51, 61, 71, 81, 91, 22, 32, 42

Offset: 0

Reinhard Zumkeller, Nov 03 2014

a(n) = A102823(n) for n <= 55;
not all numbers occur: all repunits (A002275) greater than 1 are missing; idea of proof: for n > 1 the digit 1 will never again be the smallest of least frequently occurring digits;
A249648 gives positions of terms containing a zero.

			n = 11: digits 0 and 1 occur twice in {a(k): k=0..10}, all other digits exactly once, where 2 is the smallest; therefore a(11) must contain digit 2, and 12 is the smallest unused number containing 2, hence a(11) = 12.
n = 55: digits 0..9 occur exactly 10 times in {a(k): k=0..54}; therefore a(55) must contain digit 0, the smallest digit; a(55) = 100, as 100 is the smallest unused number containing 0;
n = 56: least occurring digits in {a(k): k=0..10} are 2..9 and 2 is the smallest; therefore a(56) must contain digit 2, and 21 is the smallest unused number containing 2, hence a(56) = 21.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A031298, A002275, A011540, A249648, A102823.

import Data.List (delete, group, sortBy); import Data.Function (on)
a249626 n = a249626_list !! n
a249626_list = f (zip [0,0..] [0..9]) a031298_tabf where
   f acds@((,dig):) zss = g zss where
     g (ys:yss) = if dig `elem` ys
                     then y : f acds' (delete ys zss) else g yss
       where y = foldr (\d v -> 10 * v + d) 0 ys
             acds' = sortBy (compare `on` fst) $
                    addd (sortBy (compare `on` snd) acds)
                         (sortBy (compare `on` snd) $
                                 zip (map length gss) (map head gss))
             addd cds [] = cds
             addd []   _ = []
             addd ((c, d) : cds) yys'@((cy, dy) : yys)
                  | d == dy  = (c + cy, d) : addd cds yys
                  | otherwise = (c, d) : addd cds yys'
             gss = sortBy compare $ group ys

A251984 Smallest number such that a carry occurs when adding it to n in decimal representation.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 90, 9, 8, 7, 6, 5, 4, 3, 2, 1, 80, 9, 8, 7, 6, 5, 4, 3, 2, 1, 70, 9, 8, 7, 6, 5, 4, 3, 2, 1, 60, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 9, 8, 7, 6

Offset: 1

Reinhard Zumkeller, Dec 12 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Carry
Wikipedia, Carry (arithmetic)

Cf. A004151, A122840, A031298, A167832.

a251984 n = if d > 0 then 10 - d else 10 * a251984 n'
            where (n',d) = divMod n 10

def a(n):
    s = str(n)
    t = s.strip('0')
    return (10 - int(t)%10) * 10**(len(s) - len(t))
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Sep 08 2021

a(n) = (10 - A004151(n) mod 10) * 10^A122840(n).

A257831 In decimal representation of n: replace each digit with its binary representation.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 10, 11, 110, 111, 1100, 1101, 1110, 1111, 11000, 11001, 100, 101, 1010, 1011, 10100, 10101, 10110, 10111, 101000, 101001, 110, 111, 1110, 1111, 11100, 11101, 11110, 11111, 111000, 111001, 1000, 1001, 10010, 10011

Offset: 0

Reinhard Zumkeller, May 10 2015

a(n) = n iff (largest digit of n) = 1: a(A007088(n)) = A007088(n);

a(n) = A007088(A080719(n)).

			.    n |  dec    -->  bin         |     a(n) |  A080719(n)
. -----+--------------------------+----------+------------
.  100 |  [1,0,0]  |  [1,0,0]     |     100  |          4
.  101 |  [1,0,1]  |  [1,0,1]     |     101  |          5
.  102 |  [1,0,2]  |  [1,0,10]    |    1010  |         10
.  103 |  [1,0,3]  |  [1,0,11]    |    1011  |         11
.  104 |  [1,0,4]  |  [1,0,100]   |   10100  |         20
.  105 |  [1,0,5]  |  [1,0,101]   |   10101  |         21
.  106 |  [1,0,6]  |  [1,0,110]   |   10110  |         22
.  107 |  [1,0,7]  |  [1,0,111]   |   10111  |         23
.  108 |  [1,0,8]  |  [1,0,1000]  |  101000  |         40
.  109 |  [1,0,9]  |  [1,0,1001]  |  101001  |         41
.  110 |  [1,1,0]  |  [1,1,0]     |     110  |          6
.  111 |  [1,1,1]  |  [1,1,1]     |     111  |          7
.  112 |  [1,1,2]  |  [1,1,10]    |    1110  |         14
.  113 |  [1,1,3]  |  [1,1,11]    |    1111  |         15
.  114 |  [1,1,4]  |  [1,1,100]   |   11100  |         28
.  115 |  [1,1,5]  |  [1,1,101]   |   11101  |         29
.  116 |  [1,1,6]  |  [1,1,110]   |   11110  |         30
.  117 |  [1,1,7]  |  [1,1,111]   |   11111  |         31
.  118 |  [1,1,8]  |  [1,1,1000]  |  111000  |         56
.  119 |  [1,1,9]  |  [1,1,1001]  |  111001  |         57
.  120 |  [1,2,0]  |  [1,10,0]    |    1100  |         12
.  121 |  [1,2,1]  |  [1,10,1]    |    1101  |         13
.  122 |  [1,2,2]  |  [1,10,10]   |   11010  |         26
.  123 |  [1,2,3]  |  [1,10,11]   |   11011  |         27
.  124 |  [1,2,4]  |  [1,10,100]  |  110100  |         52
.  125 |  [1,2,5]  |  [1,10,101]  |  110101  |         53  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088, A030308, A031298, A080719.

import Data.Maybe (mapMaybe)
a257831 = foldr (\b v -> 10 * v + b) 0 .
           concat . mapMaybe (flip lookup bin) . a031298_row
            where bin = zip [0..9] a030308_tabf

Table[FromDigits[Flatten[IntegerDigits[#,2]&/@IntegerDigits[n]]],{n,0,50}] (* Harvey P. Dale, Jun 06 2020 *)

def A257831(n):
    return int(''.join((format(int(d),'b') for d in str(n))))
# Chai Wah Wu, May 10 2015

A265525 a(n) = largest base-10 palindrome m <= n such that every base-10 digit of m is <= the corresponding digit of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 11, 22, 22, 22, 22, 22, 22, 22, 22, 0, 11, 22, 33, 33, 33, 33, 33, 33, 33, 0, 11, 22, 33, 44, 44, 44, 44, 44, 44, 0, 11, 22, 33, 44, 55, 55, 55, 55, 55, 0, 11, 22, 33, 44, 55, 66, 66, 66, 66, 0, 11, 22, 33, 44, 55, 66, 77, 77, 77, 0, 11, 22, 33

Offset: 0

N. J. A. Sloane, Dec 09 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Cf. A002113, A031298, A265558.

a265525 n = a265525_list !! n
a265525_list = f a031298_tabf [[]] where
   f (ds:dss) pss = y : f dss pss' where
     y = foldr (\d v -> 10 * v + d) 0 ys
     (ys:_) = dropWhile (\ps -> not $ and $ zipWith (<=) ps ds) pss'
     pss' = if ds /= reverse ds then pss else ds : pss
-- Reinhard Zumkeller, Dec 11 2015

ispal := proc(n) # test for base-b palindrome
local L, Ln, i;
global b;
L := convert(n, base, b);
Ln := nops(L);
for i to floor(1/2*Ln) do
if L[i] <> L[Ln + 1 - i] then return false end if
end do;
return true
end proc
# find max pal <= n and in base-b shadow of n, write in base 10
under10:=proc(n) global b;
local t1,t2,i,m,sw1,L2;
if n mod b = 0 then return(0); fi;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then
t2:=convert(m,base,b);
L2:=nops(t2);
sw1:=1;
for i from 1 to L2 do
if t2[i] > t1[i] then sw1:=-1; break; fi;
od:
if sw1=1 then return(m); fi;
fi;
od;
end proc;
b:=10; [seq(under10(n),n=0..144)]; # Gives A265525

cp's OEIS Frontend

A261793 Successively add the smallest number that is not a substring in decimal representation.

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A080719 Replace decimal digits with their binary values and convert back to decimal representation.

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A261794 a(n) is the smallest nonzero number that is not a substring of n in decimal representation.

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A262437 Triangle T(n,k): write n in base 16, reverse order of digits.

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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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A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

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A249626 a(0) = 0, a(n+1) = smallest number, not occurring earlier, containing the smallest of the least frequently occurring digits in all preceding terms.

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A251984 Smallest number such that a carry occurs when adding it to n in decimal representation.

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