A047813 -id:A047813 - cp's OEIS frontend

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

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

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Length of row n = A262190(n);
T(n,0) = A054054(n);
T(n,A262190(n)-1) = A047813(n).

			.     n |  T(n,*)           n |  T(n,*)              n |  T(n,*)
.  -----+-----------    ------+-------------    -------+--------------
.   100 |  0,1           1000 |  0,1             10000 |  0,1
.   101 |  0,1,101       1001 |  0,1,1001        10001 |  0,1,10001
.   102 |  0,1,2         1002 |  0,1,2           10002 |  0,1,2
.   103 |  0,1,3         1003 |  0,1,3           10003 |  0,1,3
.   104 |  0,1,4         1004 |  0,1,4           10004 |  0,1,4
.   105 |  0,1,5         1005 |  0,1,5           10005 |  0,1,5
.   106 |  0,1,6         1006 |  0,1,6           10006 |  0,1,6
.   107 |  0,1,7         1007 |  0,1,7           10007 |  0,1,7
.   108 |  0,1,8         1008 |  0,1,8           10008 |  0,1,8
.   109 |  0,1,9         1009 |  0,1,9           10009 |  0,1,9
.   110 |  0,1,11        1010 |  0,1,101         10010 |  0,1,1001
.   111 |  1,11,111      1011 |  0,1,11,101      10011 |  0,1,11,1001
.   112 |  1,2,11        1012 |  0,1,2,101       10012 |  0,1,2,1001
.   113 |  1,3,11        1013 |  0,1,3,101       10013 |  0,1,3,1001
.   114 |  1,4,11        1014 |  0,1,4,101       10014 |  0,1,4,1001
.   115 |  1,5,11        1015 |  0,1,5,101       10015 |  0,1,5,1001
.   116 |  1,6,11        1016 |  0,1,6,101       10016 |  0,1,6,1001
.   117 |  1,7,11        1017 |  0,1,7,101       10017 |  0,1,7,1001
.   118 |  1,8,11        1018 |  0,1,8,101       10018 |  0,1,8,1001
.   119 |  1,9,11        1019 |  0,1,9,101       10019 |  0,1,9,1001
.   120 |  0,1,2         1020 |  0,1,2           10020 |  0,1,2
.   121 |  1,2,121       1021 |  0,1,2           10021 |  0,1,2
.   122 |  1,2,22        1022 |  0,1,2,22        10022 |  0,1,2,22
.   123 |  1,2,3         1023 |  0,1,2,3         10023 |  0,1,2,3
.   124 |  1,2,4         1024 |  0,1,2,4         10024 |  0,1,2,4
.   125 |  1,2,5         1025 |  0,1,2,5         10025 |  0,1,2,5  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Index entries for sequences related to palindromes

Cf. A262190 (row lengths), A054054 (left edge), A047813 (right edge), A136522, A002113.

Cf. A031298, A218978.

import Data.List (inits, tails, nub, sort)
a262188 n k = a262188_tabf !! n !! k
a262188_row n = a262188_tabf !! n
a262188_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
   filter (\xs -> length xs == 1 || last xs > 0 && reverse xs == xs) .
          concatMap (tail . inits) . tails) a031298_tabf

A262188_row(n,b=10)=Set(concat(vector(logint(n+!n,b)+1,m,m=n\=b^(m>1);select(is_A002113,vector(logint(m+!m,b)+1,k,m%b^k))))) \\ M. F. Hasler, Jun 19 2018

A262224 a(n+1) = a(n) + (largest palindrome in decimal representation of a(n)), a(0) = 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 22, 44, 88, 176, 183, 191, 382, 390, 399, 498, 507, 514, 519, 528, 536, 542, 547, 554, 609, 618, 626, 1252, 1504, 1509, 1518, 1669, 1735, 1742, 1749, 1758, 1766, 1832, 1840, 1848, 2696, 3392, 3425, 3430, 3773, 7546, 7553, 7608, 7616, 8232

Offset: 0

Reinhard Zumkeller, Sep 15 2015

a(n+1) = a(n) + A047813(a(n)) = A262223(a(n)).

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

Cf. A047813, A262223, A262243 (first differences).

a262224' n = a262224_list' !! n
a262224_list' = iterate a262223 1

A046260 Largest palindromic substring in 2^n.

Original entry on oeis.org

1, 2, 4, 8, 6, 3, 6, 8, 6, 5, 4, 8, 9, 9, 8, 8, 55, 131, 262, 242, 8, 9, 9, 838, 777, 55, 88, 77, 545, 9, 737, 474, 949, 858, 717, 383, 767, 9, 77, 888, 777, 255552, 111, 222, 444, 888, 77, 3553, 767, 21312, 42624, 99, 737, 474, 9, 797, 575, 8558, 7117, 646, 606, 939

Offset: 0

Patrick De Geest, Jun 15 1998

			2^41 = 2199023{255552}.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A000079, A047813, A046268.

def c(s): return s[0] != "0" and s == s[::-1]
def a(n):
    s = str(2**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(62)]) # Michael S. Branicky, Sep 18 2022

a(n) = A047813(A000079(n)). - Michel Marcus, Sep 19 2022

A262223 a(n) = n + largest palindrome contained in decimal representation of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 14, 16, 18, 20, 22, 24, 26, 28, 22, 23, 44, 26, 28, 30, 32, 34, 36, 38, 33, 34, 35, 66, 38, 40, 42, 44, 46, 48, 44, 45, 46, 47, 88, 50, 52, 54, 56, 58, 55, 56, 57, 58, 59, 110, 62, 64, 66, 68, 66, 67, 68, 69, 70, 71

Offset: 0

Reinhard Zumkeller, Sep 15 2015

a(n) = n + A047813(n).

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

Cf. A047813, A262224 (iteration, start = 1).

Haskell
```
a262223 n = n + a047813 n
```

A262243 First differences of A262224.

Original entry on oeis.org

1, 2, 4, 8, 6, 22, 44, 88, 7, 8, 191, 8, 9, 99, 9, 7, 5, 9, 8, 6, 5, 7, 55, 9, 8, 626, 252, 5, 9, 151, 66, 7, 7, 9, 8, 66, 8, 8, 848, 696, 33, 5, 343, 3773, 7, 55, 8, 616, 232, 464, 9, 9, 9, 55, 9, 9, 9, 9, 9, 55, 11, 121, 242, 484, 99, 7, 7, 8, 9, 9, 101, 8

Offset: 0

Reinhard Zumkeller, Sep 15 2015

All terms are palindromic by definition of A262224.

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

Cf. A262224, A047813.

a262243 n = a262243_list !! n
a262243_list = zipWith (-) (tail a262224_list) a262224_list

a(n) = A262224(n+1) - A262224(n) = A047813(A262224(n)).

A184988 Largest palindromic substring of n^2.

Original entry on oeis.org

0, 1, 4, 9, 6, 5, 6, 9, 6, 8, 1, 121, 44, 9, 9, 22, 6, 9, 4, 6, 4, 44, 484, 9, 7, 6, 676, 9, 8, 8, 9, 9, 4, 9, 11, 22, 9, 9, 444, 5, 6, 8, 7, 9, 9, 202, 11, 22, 4, 4, 5, 6, 7, 9, 9, 5, 313, 9, 33, 8, 6, 7, 44, 969, 9, 22, 6, 44, 6, 7, 9, 5, 8, 9, 7, 6, 77, 929

Offset: 0

Jonathan Vos Post, Mar 27 2011

Leading 0's are not allowed, so a(103) = 9, not 060. - Robert Israel, Nov 05 2020

			a(15) = largest palindromic substring of 15^2 = largest palindromic substring of 225 = 22.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000290, A002113, A047813.

g:= proc(S) S[1] <> "0" and S = StringTools:-Reverse(S) end proc:
f:= proc(n) local S,d,l,i,Q;
  S:= sprintf("%d",n^2);
  d:= length(S);
  for l from d to 1 by -1 do
    Q:= select(g, [seq(S[i..i+l-1],i=1..d+1-l)]);
    if Q <> [] then return op(sscanf(max(Q),"%d")) fi
  od;
end proc:
f(0):= 0:
map(f, [$0..200]); # Robert Israel, Nov 05 2020

a(n) = A047813(A000290(n)).

A331804 a(n) is the largest positive integer occurring, when written in binary, as a substring in both binary n and its reversal (A030101(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 3, 5, 7, 15, 1, 17, 9, 9, 5, 21, 6, 7, 3, 9, 5, 27, 7, 7, 15, 31, 1, 33, 17, 17, 9, 9, 9, 9, 5, 9, 21, 21, 6, 45, 14, 15, 3, 17, 9, 51, 5, 21, 27, 27, 7, 9, 7, 27, 15, 15, 31, 63, 1, 65, 33, 33, 17, 17, 17, 17, 9, 73, 10, 9

Offset: 0

Rémy Sigrist, Jan 26 2020

We set a(0) = 0 by convention.

a(7479) = 29 ("11101" in binary) is the first term that does not belong to A057890.

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     3      11         11
   4     1     100          1
   5     5     101        101
   6     3     110         11
   7     7     111        111
   8     1    1000          1
   9     9    1001       1001
  10     5    1010        101
  11     5    1011        101
  12     3    1100         11

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A006995, A030101, A047813, A057890, A175466.

sub(n) = { my (b=binary(n), s=[0]); for (i=1, #b, if (b[i], for (j=i, #b, s=setunion(s, Set(fromdigits(b[i..j], 2)))))); return (s) }
a(n) = my (i=setintersect(sub(n), sub(fromdigits(Vecrev(binary(n)),2)))); i[#i]

a(n) = A175466(n, A030101(n)) for any n > 0.

a(n) <= n with equality iff n is a binary palindrome (A006995).

cp's OEIS Frontend

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

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A262224 a(n+1) = a(n) + (largest palindrome in decimal representation of a(n)), a(0) = 1.

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A046260 Largest palindromic substring in 2^n.

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Python

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A262223 a(n) = n + largest palindrome contained in decimal representation of n.

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Haskell

A262243 First differences of A262224.

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Haskell

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A184988 Largest palindromic substring of n^2.

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Maple

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A331804 a(n) is the largest positive integer occurring, when written in binary, as a substring in both binary n and its reversal (A030101(n)).

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