A262188 -id:A262188 - cp's OEIS frontend

A054054 Smallest digit of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Apr 29 2000

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, i.e., except for a set of density 0: As the number of digits of n grows, the probability of having no zero digit goes to zero as 0.9^(length of n), although there are infinitely many counterexamples. - M. F. Hasler, Oct 11 2015

			a(12) = 1 because 1 < 2.

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

Cf. A054055.

Cf. A061601, A011540, A052382, A262188.

a054054 = f 9 where
   f m x | x <= 9 = min m x
         | otherwise = f (min m d) x' where (x',d) = divMod x 10
-- Reinhard Zumkeller, Jun 20 2012, Apr 25 2012

seq(min(convert(n,base,10)),n=0..100); # Robert Israel, Jul 07 2016

Mathematica
```
A054054[n_]:=Min[IntegerDigits[n]]
```

A054054(n)=if(n,vecmin(digits(n)))  \\ or: Set(digits(n))[1]. - M. F. Hasler, Jan 23 2013

a(A011540(n)) = 0; a(A052382(n)) > 0. - Reinhard Zumkeller, Apr 25 2012
a(n) = A262188(n,0). - Reinhard Zumkeller, Sep 14 2015
a(n) = 0 iff A007954(n) = 0. - M. F. Hasler, Oct 11 2015
a(n) = 9 - A054055(A061601(n)). - Robert Israel, Jul 07 2016

Edited by M. F. Hasler, Oct 11 2015

A047813 Largest palindromic substring of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 22, 3, 4, 5, 6, 7, 8, 9, 3, 3, 3, 33, 4, 5, 6, 7, 8, 9, 4, 4, 4, 4, 44, 5, 6, 7, 8, 9, 5, 5, 5, 5, 5, 55, 6, 7, 8, 9, 6, 6, 6, 6, 6, 6, 66, 7, 8, 9, 7, 7, 7, 7, 7, 7, 7, 77, 8, 9, 8, 8, 8, 8, 8, 8, 8, 8, 88, 9, 9, 9, 9, 9, 9, 9

Offset: 0

N. J. A. Sloane

a(n) = A262188(n,A262190(n)-1). - Reinhard Zumkeller, Sep 14 2015

			a(1313) = Max{1,3,131,313} = 313.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Cf. A136522.

Cf. A262188, A262223, A262224.

a047813 = last . a262188_row
-- Reinhard Zumkeller, Sep 14 2015, Aug 23 2011

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; f[n_] := Block[{id = IntegerDigits@ n, mx = -Infinity}, k = Length@ id; While[k > 0 && mx == -Infinity, mx = Max[mx, Select[ FromDigits@# & /@ Partition[id, k, 1], palQ[#, 10] &]]; k--]; mx] (* Robert G. Wilson v, Aug 24 2011 *)
lps[n_]:=Module[{idn=IntegerDigits[n]},Max[FromDigits/@Select[ Flatten[ Table[ Partition[ idn,i,1],{i,Length[idn]}],1],#==Reverse[#]&]]]; Array[ lps,100,0] (* Harvey P. Dale, Jan 09 2015 *)

def c(s): return (s[0] != "0" or s == "0") and s == s[::-1]
def a(n):
    s = str(n)
    ss = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1))
    return max(int(w) for w in ss if c(w))
print([a(n) for n in range(96)]) # Michael S. Branicky, Sep 18 2022

A218978 Table read by rows: n-th row lists all distinct substrings of decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 10, 1, 11, 1, 2, 12, 1, 3, 13, 1, 4, 14, 1, 5, 15, 1, 6, 16, 1, 7, 17, 1, 8, 18, 1, 9, 19, 0, 2, 20, 1, 2, 21, 2, 22, 2, 3, 23, 2, 4, 24, 2, 5, 25, 2, 6, 26, 2, 7, 27, 2, 8, 28, 2, 9, 29, 0, 3, 30, 1, 3, 31, 2, 3, 32, 3

Offset: 0

Reinhard Zumkeller, May 02 2015, Nov 10 2012

A120004(n) = length of n-th row;

A154771(n) = sum of n-th row.

			Rows 100 .. 112:
.  100:  {0, 1, 10, 100},
.  101:  {0, 1, 10, 101},
.  102:  {0, 1, 2, 10, 102},
.  103:  {0, 1, 3, 10, 103},
.  104:  {0, 1, 4, 10, 104},
.  105:  {0, 1, 5, 10, 105},
.  106:  {0, 1, 6, 10, 106},
.  107:  {0, 1, 7, 10, 107},
.  108:  {0, 1, 8, 10, 108},
.  109:  {0, 1, 9, 10, 109},
.  110:  {0, 1, 10 ,11, 110},
.  111:  {1, 11, 111},
.  112:  {1, 2, 11, 12, 112}.

Reinhard Zumkeller, Rows n=0..1000 of triangle, flattened

Cf. A031298, A219031 (squares in row), A262188 (palindromes in row).

import Data.List (inits, tails, sort, nub, genericIndex)
a218978 n k = a218978_row n !! k
a218978_row n = genericIndex a218978_tabf n
a218978_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
                   concatMap (tail . inits) . tails) a031298_tabf
-- Reinhard Zumkeller, corrected: Sep 15 2015, May 02 2015, Nov 10 2012

A262190 Number of distinct palindromes contained as substrings in the decimal representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Length of row n in table A262188.

Cf. A055642, A262198.

Haskell
```
a262190 = length . a262188_row
```

A262190(n)=#A262188_row(n) \\ M. F. Hasler, Jun 19 2018

a(n) = A055642(n) for n < 100 = A262198(1);

a(n) != A055642(n) iff n is in A262198.

Edited by M. F. Hasler, Jun 19 2018

A262198 Numbers such that the number of distinct palindromes contained as substring in their decimal representation differs from the length thereof.

Original entry on oeis.org

100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1020, 1021, 1030, 1031, 1040, 1041, 1050, 1051, 1060, 1061, 1070, 1071, 1080, 1081, 1090, 1091, 1100, 1200, 1201, 1231, 1241, 1251, 1261, 1271

Offset: 1

Reinhard Zumkeller, Sep 14 2015

Or, numbers n such that A055642(n) != A262190(n).

a(22) = 1021 is the first term which differs from "numbers having at least two digits 0 in their decimal representation" (not in OEIS). It seems that A043490 is a subsequence. - M. F. Hasler, Jun 19 2018

			a(39) = 1201, containing just 3 trivial palindromes 0, 1 and 2;
1202, also of length = 4, contains exactly 4 palindromes 0, 1, 2 and 202, therefore 1202 is not a term.

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

Cf. A055642, A262190, A262188.

a262198 n = a262198_list !! (n-1)
a262198_list = [x | x <- [0..], a055642 x /= a262190 x]

is(n)=#digits(n)!=A262190(n) \\ M. F. Hasler, Jun 19 2018

Definition clarified by M. F. Hasler, Jun 19 2018

cp's OEIS Frontend

A054054 Smallest digit of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A047813 Largest palindromic substring of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A218978 Table read by rows: n-th row lists all distinct substrings of decimal representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A262190 Number of distinct palindromes contained as substrings in the decimal representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

PARI

Formula

Extensions

A262198 Numbers such that the number of distinct palindromes contained as substring in their decimal representation differs from the length thereof.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Extensions