A260255 -id:A260255 - cp's OEIS frontend

A262528 Maximum number of backward steps k needed to find a representation of an n-digit decimal number x as a sum of three base-10 palindromes of the form k-th largest base-10 palindrome <= x plus a number representable as sum of two base-10 palindromes from A260255.

0, 1, 1, 3, 3, 11, 4, 10, 4, 23, 9, 15, 6, 23, 11

Offset: 1

Hugo Pfoertner, Sep 26 2015

The sequence terms are counterexamples to the second part of the claim stated in the answers to the Math Magic Problem of the Month (June 1999) that "all sufficiently large numbers seem to be the sum of 3 palindromes, one of which is the biggest or second biggest possible", which would mean all a(k)=2 for k "sufficiently large".
Since exhaustive search is currently (2015) considered as not feasible, a(16)>=16, a(17)>=7, a(18)>=25, a(19)>=14 are only lower bounds for the next sequence terms.
M. Sigg has shown that a(n)>=3 for n = 5 + 4 * j.

			a(1)=0 because all 1-digit numbers are palindromes,
a(2)=a(3)=1 because all 2-digit and all 3-digit numbers can be represented by the nearest smaller palindrome and a number <=10, e.g., 201=191+9+1.
a(4)=3, because for the number 2023 the largest palindrome leading to a difference representable as sum of two palindromes is 1881. 2023-2002=21 and 2023-1991=32 are not in A260255. 2023-1881=142=141+1 is in A260255. No other 4-digit number requires more than 3 backward steps.
a(6)=11 because for the 6-digit number 101199 none of the first 10 differences 101199-101101=98, 101199-10001=1198, 101199-99999=1200, 101199-99899=1300, 101199-99799=1400, 101199-99699=1500, 101199-99599=1600, 101199-99499=1700, 101199-99399=1800, 101199-99299=1900 is representable as sum of two palindromes (i.e., are in A035137), whereas the 11th palindrome 99199 leads to 101199-99199=2000=1991+9.
a(18)>=25 because for the number x=100000001814566071 only the 25th palindrome < x 99999997779999999 produces the first difference 4034566072 representable as sum of 2 palindromes.

Erich Friedman, Problem of the Month (June 1999)
Markus Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, arXiv:1510.07507 [math.NT], 2015.

Cf. A002113, A035137, A088601, A109326, A260255, A261132, A261675.

A035137 Numbers that are not the sum of 2 palindromes (where 0 is considered a palindrome).

Original entry on oeis.org

21, 32, 43, 54, 65, 76, 87, 98, 201, 1031, 1041, 1042, 1051, 1052, 1053, 1061, 1062, 1063, 1064, 1071, 1072, 1073, 1074, 1075, 1081, 1082, 1083, 1084, 1085, 1086, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1099, 1101, 1103, 1104, 1105, 1106, 1107, 1108

Offset: 1

Patrick De Geest, Nov 15 1998

Apparently, every positive number is equal to the sum of at most 3 positive palindromes. - Giovanni Resta, May 12 2013
A260254(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2015
A261675(a(n)) >= 3 (and, conjecturally, = 3). - N. J. A. Sloane, Sep 03 2015
This sequence is infinite. Proof: It is easy to see that 200...01 (with any number of zeros) cannot be the sum of two palindromes. - N. J. A. Sloane, Sep 03 2015
The conjecture that every number is the sum of 3 palindromes fails iff there is a term a(n) such that for all palindromes P < a(n), the difference a(n) - P is also a term of this sequence. - M. F. Hasler, Sep 08 2015
Cilleruelo and Luca (see links) have proved the conjecture that every positive integer is the sum of at most three palindromes (in bases >= 5), and also that the density of those that require three is positive. - Christopher E. Thompson, Apr 14 2016

David W. Wilson, Table of n, a(n) for n = 1..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv:1602.06208 [math.NT], 2016.
P. De Geest, World!Of Numbers
Hugo Pfoertner, Plot of first 10^6 terms
Eric Weisstein's World of Mathematics, Palindromic Number

Cf. A014091, A014092, A104444.
Cf. A260254, A260255 (complement), A002113, A261906, A261907.
Cf. A319477 (disallowing zero).

a035137 n = a035137_list !! (n-1)
a035137_list = filter ((== 0) . a260254) [0..]
-- Reinhard Zumkeller, Jul 21 2015

N:= 4: # to get all terms with <= N digits
revdigs:= proc(n) local L,j,nL;
  L:= convert(n,base,10); nL:= nops(L);
  add(L[j]*10^(nL-j),j=1..nL);
end proc;
palis:= $0..9:
for d from 2 to N do
  if d::even then
    palis:= palis, seq(x*10^(d/2)+revdigs(x),x=10^(d/2-1)..10^(d/2)-1)
  else
    palis:= palis, seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+revdigs(x),y=0..9),x=10^((d-3)/2)..10^((d-1)/2)-1);
  fi
od:
palis:= [palis]:
A:= Array(0..10^N-1):
A[palis]:= 1:
B:= SignalProcessing:-Convolution(A,A):
select(t -> B[t+1] < 0.5, [$1..10^N-1]); # Robert Israel, Jun 22 2015

palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; nn=1108; t={}; Do[i=c=0; While[i<=n && c!=1,If[palQ[i] && palQ[n-i], AppendTo[t,n]; c=1]; i++],{n,nn}]; Complement[Range[nn],t] (* Jayanta Basu, May 12 2013 *)

is_A035137(n)={my(k=0);!until(n<2*k=nxt(k),is_A002113(n-k)&&return)} \\ Uses function nxt() given in A002113. Not very efficient for large n, better start with k=n-A261423(n). Maybe also better use A261423 rather than nxt(). - M. F. Hasler, Jul 21 2015

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)}.

A261675 Minimal number of palindromes in base 10 that add to n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 02 2015

This sequence coincides with A088601 for n <= 301, but differs at n=302.
Although A088601 and this sequence agree for a large number of terms, because of their importance they warrant separate entries.
Cilleruelo and Luca prove that a(n) <= 3 (in fact they prove this for any fixed base g>=5). - Danny Rorabaugh, Feb 26 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv preprint arXiv:1602.06208 [math.NT], 2016.
William D. Banks, Every natural number is the sum of forty-nine palindromes, INTEGERS 17 (2016), 9 pp.
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes (2018), Numberphile video

Cf. A002113, A035137, A088601, A260255, A261422.

ispal(n)=my(d=digits(n)); d==Vecrev(d);
a(n)=my(L=n\2,d,e); if(ispal(n), return(1)); d=[1]; while((e=fromdigits(d))<=L, if(ispal(n-e), return(2)); my(k=#d,i=(k+1)\2); while(i&&d[i]==9, d[i]=0; d[k+1-i]=0; i--); if(i, d[i]++; d[k+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1)); 3; \\ Charles R Greathouse IV, Nov 12 2018

A261906 Numbers that are the sum of two nonzero palindromes.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

N. J. A. Sloane, Sep 09 2015

More than the usual number of terms are shown in order to distinguish this from A260255.

			22 is a member because it is the sum of two palindromes, 11+11 (not because it is a palindrome in its own right).
111 is not the sum of two nonzero palindromes, so appears in A260255 but not here. See A213879 for further differences between the two sequences.

Cf. A002113, A260255, A213879.

# Sums of two nonzero pals:
# bP has a list of palindromes starting at 0.
a2:={}; M:=60; M2:=bP[M];
for i from 2 to M do
for j from i to M do
k:=bP[i]+bP[j];
if k <= M2 then a2:={op(a2),k}; fi;
od: od:
b2:=sort(convert(a2,list));

Take[Total/@Tuples[Select[Range[200],PalindromeQ],2]//Union,120] (* Harvey P. Dale, Apr 27 2025 *)

A261907 Numbers that are the sum of two nonzero palindromes but are not palindromes themselves.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112

Offset: 1

N. J. A. Sloane, Sep 09 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..7814

Equals A261906 \ A002113. Cf. A213879, A260255.

# bP has a list of all palindromes (from A002113):
a3:={}; M:=60; M2:=bP[M];
for i from 2 to M do
for j from i to M do
k:=bP[i]+bP[j];
if k <= M2 and digrev(k) <> k then a3:={op(a3),k}; fi;
od: od:
b3:=sort(convert(a3,list));

A213879 Positive palindromes that are not the sum of two positive palindromes.

Original entry on oeis.org

1, 111, 131, 141, 151, 161, 171, 181, 191, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10301, 10401, 10501, 10601, 10701, 10801, 10901, 11111, 11211, 11311, 11411, 11511, 11611, 11711, 11811, 11911, 12021, 12121, 12321, 12421, 12521, 12621, 12721, 12821

Offset: 1

Arkadiusz Wesolowski, Jun 23 2012

These numbers do not occur in A035137.

			22 is not a member because 22 = 11 + 11.

Alois P. Heinz, Table of n, a(n) for n = 1..2151 (first 111 terms from N. J. A. Sloane)
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A002113, A014092, A035137, A083142, A088601, A260255, A319477.

# From N. J. A. Sloane, Sep 09 2015: bP is a list of the palindromes
a:={}; M:=400; for n from 3 to M do p:=bP[n];
# is p a sum of two palindromes?
sw:=-1; for i from 2 to n-1 do j:=p-bP[i]; if digrev(j)=j then sw:=1; break; fi;
od;
if sw<0 then a:={op(a),p}; fi; od:
b:=sort(convert(a,list));

lst1 = {}; lst2 = {}; r = 12821; Do[If[FromDigits@Reverse@IntegerDigits[n] == n, AppendTo[lst1, n]], {n, r}]; l = Length[lst1]; Do[s = lst1[[i]] + lst1[[j]]; AppendTo[lst2, s], {i, l - 1}, {j, i}]; Complement[lst1, lst2]
palQ[n_] := Reverse[x = IntegerDigits[n]] == x; t1 = Select[Range[12900], palQ[#] &]; Complement[t1, Union[Flatten[Table[i + j, {i, t1}, {j, t1}]]]] (* Jayanta Basu, Jun 15 2013 *)

({ A002113 } intersect { A319477 }) minus { 0 }. - Alois P. Heinz, Sep 19 2018

A318535 Integers a(n) such that [a(n) - k] is a palindrome in base 10, with k <10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 28 2018

This is not the sequence A260255 although more than the first 111 terms are equal.

			21 is not in the sequence because 11 (the closest palindromic integer < 21) is not reachable by subtracting an integer < 10 from 21. The integers 32, 43 or 2018 are not in the sequence for the same reason.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10868

Cf. A318536 for an equivalent sequence [addition of (k < 10) instead of subtraction].

Cf. A261906.

A263994 First element of first run of at least n consecutive numbers not representable as the sum of two base-10 palindromes.

Original entry on oeis.org

21, 1041, 1051, 1061, 1071, 1081, 1091, 107209, 108429, 109803, 10715097, 10854691, 10904639, 10904731, 10904731, 10904731, 10904731, 10983831, 11002311, 11002311, 11002311, 1078112970, 1078122970

Offset: 1

Hugo Pfoertner, Oct 31 2015

First occurrence of n consecutive numbers in A035137.

			a(2)=1041 because 1041 and 1042 are the first two consecutive numbers in A035137.

Hugo Pfoertner, Table of n, a(n) for n = 1..52. Checked up to 2*10^11.

Cf. A002113, A035137, A260255.

A287961 Numbers that are the sum of two palindromic primes (A002385).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 22, 103, 104, 106, 108, 112, 133, 134, 136, 138, 142, 153, 154, 156, 158, 162, 183, 184, 186, 188, 192, 193, 194, 196, 198, 202, 232, 252, 262, 282, 292, 302, 312, 315, 316, 318, 320, 322, 324, 332, 342, 355, 356, 358, 360, 362, 364, 372, 375, 376, 378, 380

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

Eric Weisstein's World of Mathematics, Palindromic Prime
Index entries for sequences related to palindromes

Cf. A002113, A002385, A014091, A035137, A260254, A260255, A261906.

nmax = 380; f[x_] := Sum[Boole[PalindromeQ[k] && PrimeQ[k]] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

cp's OEIS Frontend

A262528 Maximum number of backward steps k needed to find a representation of an n-digit decimal number x as a sum of three base-10 palindromes of the form k-th largest base-10 palindrome <= x plus a number representable as sum of two base-10 palindromes from A260255.

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A035137 Numbers that are not the sum of 2 palindromes (where 0 is considered 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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A261675 Minimal number of palindromes in base 10 that add to n.

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A261906 Numbers that are the sum of two nonzero palindromes.

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A261907 Numbers that are the sum of two nonzero palindromes but are not palindromes themselves.

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A213879 Positive palindromes that are not the sum of two positive palindromes.

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A318535 Integers a(n) such that [a(n) - k] is a palindrome in base 10, with k <10.

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A263994 First element of first run of at least n consecutive numbers not representable as the sum of two base-10 palindromes.