A136522 -id:A136522 - cp's OEIS frontend

A065206 Numbers which need one 'Reverse and Add' step to reach a palindrome.

10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 43, 45, 47, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 65, 70, 71, 72, 74, 80, 81, 83, 90, 92, 100, 102, 103, 104, 105, 106, 107, 108, 110, 112, 113, 114, 115, 116, 117

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (A002113) are excluded.

Numbers k such that A033665(k) = 1. - Andrew Howroyd, Dec 05 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015976, A033665, A136522, A056964, A029742.

Sequences for 2..12 steps needed are: A065207, A065208, A065209, A065210, A065211, A065212, A065213, A065214, A065215, A065216, A065217.

function revadd_steps(k,stop: integer); var c,n,m,steps,rev: integer; begin n := 0; c := 0; while c < stop do m := n; rev := int_reverse(m); steps := 0; while steps < k and m <> rev do m := m + rev; rev := int_reverse(m); inc(steps); end; if steps = k and m = rev then write(n," "); inc(c); end; inc(n); end; end; revadd_steps(1,66).

a065206 n = a065206_list !! (n-1)
a065206_list = filter ((== 1) . a136522 . a056964) a029742_list
-- Reinhard Zumkeller, Oct 14 2011

Select[Range[10,120],!PalindromeQ[#]&&PalindromeQ[#+IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2017 *)

isok(n,s=1)={for(k=0, s, my(r=fromdigits(Vecrev(digits(n)))); if(r==n, return(k==s)); n += r); 0} \\ Andrew Howroyd, Dec 05 2024

Offset corrected by Harry J. Smith, Oct 13 2009

A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Sep 19 2018

Differs from A008284 and from A072233 first at T(10,1) = 0.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  1;
  0, 1, 2,  1,  1;
  0, 1, 2,  2,  1,  1;
  0, 1, 3,  3,  2,  1,  1;
  0, 1, 3,  4,  3,  2,  1, 1;
  0, 1, 4,  5,  5,  3,  2, 1, 1;
  0, 1, 4,  7,  6,  5,  3, 2, 1, 1;
  0, 0, 5,  8,  9,  7,  5, 3, 2, 1, 1;
  0, 1, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;
  0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.
Row sums give A091580.
T(2n,n) gives A319454.
Cf. A002113, A008284, A072233, A261132.

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..14);

T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A002113(j)).

Sum_{k=0..3} T(n,k) = A261132(n).

A046447 Apart from initial term, composite numbers with the property that the concatenation of their prime factors is a palindrome.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 39, 49, 64, 69, 81, 119, 121, 125, 128, 129, 159, 219, 243, 249, 256, 259, 329, 339, 343, 403, 429, 469, 507, 512, 625, 669, 679, 729, 795, 1024, 1207, 1309, 1329, 1331, 1533, 1547, 1587, 1589, 1703, 2023, 2048, 2097, 2187, 2319

Offset: 1

Patrick De Geest, Jul 15 1998

Prime factors considered with multiplicity. - Harvey P. Dale, Apr 20 2025

			81 is a term because 81 = 3 * 3 * 3 * 3 -> 3333 is palindromic.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A046448, A046449, A046450.

Cf. A027746, A018252, A136522, A002113.

a046447 n = a046447_list !! (n-1)
a046447_list = 1 : filter f [1..] where
   f x = length ps > 1 && ps' == reverse ps'
         where ps' = concatMap show ps; ps = a027746_row x
-- Reinhard Zumkeller, May 02 2014

concat[n_]:=Flatten[Table[IntegerDigits[First[n]],{Last[n]}]]; palQ[n_]:= Module[{x=Flatten[concat/@FactorInteger[n]]},x==Reverse[x]&&!PrimeQ[n]]; Select[Range[2500],palQ] (* Harvey P. Dale, May 24 2011 *)
cpfpQ[n_]:=PalindromeQ[FromDigits[Flatten[IntegerDigits/@Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]]]]; Join[{1},Select[Range[2500],CompositeQ[ #]&&cpfpQ[#]&]] (* Harvey P. Dale, Apr 20 2025 *)

from sympy import factorint, isprime
A046447_list = [1]
for n in range(4, 10**6):
    if not isprime(n):
        s = ''.join([str(p)*e for p, e in sorted(factorint(n).items())])
        if s == s[::-1]:
            A046447_list.append(n) # Chai Wah Wu, Jan 03 2015

Definition slightly modified by Harvey P. Dale, Apr 20 2025

A260254 Number of ways to write n as sum of two palindromes in decimal representation.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Jul 21 2015

a(A035137(n)) = 0; a(A260255(n)) > 0.

			.   n | a(n) |                                n | a(n) |
. ----+------+--------------------------    ----+------+--------------
.   0 |    1 |  0                            21 |    0 |  ./.
.   1 |    1 |  1                            22 |    2 |  22, 11+11
.   2 |    2 |  2, 1+1                       23 |    1 |  22+1
.   3 |    2 |  3, 2+1                       24 |    1 |  22+2
.   4 |    3 |  4, 3+1, 2+2                  25 |    1 |  22+3
.   5 |    3 |  5, 4+1, 3+2                  26 |    1 |  22+4
.   6 |    4 |  6, 5+1, 4+2, 3+3             27 |    1 |  22+5
.   7 |    4 |  7, 6+1, 5+2, 4+3             28 |    1 |  22+6
.   8 |    5 |  8, 7+1, 6+2, 5+3, 4+4        29 |    1 |  22+7
.   9 |    5 |  9, 8+1, 7+2, 6+3, 5+4        30 |    1 |  22+8
.  10 |    5 |  9+1, 8+2, 7+3, 6+4, 5+5      31 |    1 |  22+9
.  11 |    5 |  11, 9+2, 8+3, 7+4, 6+5       32 |    0 |  ./.
.  12 |    5 |  11+1, 9+3, 8+4, 7+5, 6+6     33 |    2 |  33, 22+11
.  13 |    4 |  11+2, 9+4, 8+5, 7+6          34 |    1 |  33+1
.  14 |    4 |  11+3, 9+5, 8+6, 7+7          35 |    1 |  33+2
.  15 |    3 |  11+4, 9+6, 8+7               36 |    1 |  33+3
.  16 |    3 |  11+5, 9+7, 8+8               37 |    1 |  33+4
.  17 |    2 |  11+6, 9+8                    38 |    1 |  33+5
.  18 |    2 |  11+7, 9+9                    39 |    1 |  33+6
.  19 |    1 |  11+8                         40 |    1 |  33+7
.  20 |    1 |  11+9                         41 |    1 |  33+8  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Hugo Pfoertner, Plot of first 10^6 terms
Hugo Pfoertner, Plot of first 3*10^7 terms

Cf. A002113, A136522, A035137, A260255.

a260254 n = sum $ map (a136522 . (n -)) $
               takeWhile (<= n `div` 2) a002113_list

a(n) = sum{A136522(n - A002113(k)): k = 1..floor(n/2)}.

A064834 If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 25 2001

Might be called the Palindromic Deviation (or PD(n)) of n, since it measures how far n is from being a palindrome. - W. W. Kokko, Mar 13 2013

a(A002113(n)) = 0; a(A029742(n)) > 0; A136522(n) = A000007(a(n)). - Reinhard Zumkeller, Sep 18 2013

			a(456) = | 4 - 6 | = 2, a(4567) = | 4 - 7 | + | 5 - 6 | = 4.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Cf. A037904, A040163, A064844.

Cf. A031298, A037888.

a064834 n = sum $ take (length nds `div` 2) $
                  map abs $ zipWith (-) nds $ reverse nds
   where nds = a031298_row n
-- Reinhard Zumkeller, Sep 18 2013

f:=proc(n)
local t1,t2,i;
t1:=convert(n,base,10);
t2:=nops(t1);
add( abs(t1[i]-t1[t2+1-i]),i=1..floor(t2/2) );
end;
[seq(f(n),n=0..120)]; # N. J. A. Sloane, Mar 24 2013

f[n_] := (k = IntegerDigits[n]; l = Length[k]; Sum[ Abs[ k[[i]] - k[[l - i + 1]]], {i, 1, Floor[l/2] } ] ); Table[ f[n], {n, 0, 100} ]

from sympy import floor, ceiling
def A064834(n):
    x, y = str(n), 0
    lx2 = len(x)/2
    for a,b in zip(x[:floor(lx2)],x[:ceiling(lx2)-1:-1]):
        y += abs(int(a)-int(b))
    return y
# Chai Wah Wu, Aug 09 2014

More terms from Vladeta Jovovic, Matthew Conroy and Robert G. Wilson v, Oct 26 2001

A341155 Number of partitions of n into 2 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 3

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 3..20000

Cf. A002113, A136522, A260254, A319468, A341156, A341157, A341158, A341159.

A341156 Number of partitions of n into 3 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 9, 10, 9, 10, 8, 8, 7, 6, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 4, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 6

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 6..20000

Cf. A002113, A136522, A261131, A261132, A261422, A341155, A341157, A341158, A341159.

A341157 Number of partitions of n into 4 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 12, 13, 15, 15, 16, 16, 15, 14, 13, 11, 10, 9, 8, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 12, 12, 13, 13, 14, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 14, 16, 15, 15, 15, 15, 16

Offset: 10

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 10..20000

Cf. A002113, A136522, A319469, A341155, A341156, A341158, A341159.

A341158 Number of partitions of n into 5 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 14, 17, 17, 19, 18, 19, 17, 17, 15, 14, 12, 12, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 19, 19, 19, 20, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 21, 21, 22, 23, 24, 26, 26, 28, 28, 29, 29, 31, 29, 30, 30, 30, 30, 30, 30, 30, 30, 31

Offset: 15

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 15..20000

Cf. A002113, A136522, A319470, A341155, A341156, A341157, A341159.

A341159 Number of partitions of n into 6 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 9, 11, 13, 14, 15, 16, 16, 15, 15, 14, 13, 12, 11, 11, 11, 12, 12, 14, 15, 17, 18, 20, 20, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 22, 23, 23, 23, 24, 24, 25, 26, 28, 29, 31, 32, 35, 36, 38, 39, 40, 41, 41, 42, 42, 42, 41, 43, 42, 42, 43, 43, 44, 45, 47

Offset: 21

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 21..20000

Cf. A002113, A136522, A319471, A341155, A341156, A341157, A341158.

Table[Count[IntegerPartitions[n,{6}],?(Length[Union[#]]==Length[#]&&AllTrue[ #,PalindromeQ]&)],{n,21,100}] (* _Harvey P. Dale, Jul 27 2021 *)

cp's OEIS Frontend

A065206 Numbers which need one 'Reverse and Add' step to reach a palindrome.

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A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A046447 Apart from initial term, composite numbers with the property that the concatenation of their prime factors is a palindrome.

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A260254 Number of ways to write n as sum of two palindromes in decimal representation.

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A064834 If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.

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A341155 Number of partitions of n into 2 distinct nonzero decimal palindromes.

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A341156 Number of partitions of n into 3 distinct nonzero decimal palindromes.

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A341157 Number of partitions of n into 4 distinct nonzero decimal palindromes.

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