A035137 -id:A035137 - cp's OEIS frontend

A261908 Numbers of the form |m - R(m)| for m in A035137, where R(m) denotes the digit-reversal of m.

9, 90, 99, 180, 189, 270, 279, 360, 369, 450, 459, 540, 549, 630, 639, 720, 729, 810, 819, 900, 990, 999, 1089, 1179, 1269, 1359, 1449, 1539, 1629, 1719, 1728, 1800, 1809, 1908, 1980, 2079, 2088, 2268, 2358, 2448

Offset: 1

N. J. A. Sloane, Sep 09 2015

Created in an attempt to understand A035137. It would be nice to have an independent definition of these numbers. Obviously they are multiples of 9 - see A261909.

			21, 102, 1031 are some early terms in A035137, so this sequence contains 21-12=9, 201-102=99, 1301-1031=270.

Cf. A002113, A035137, A261909.

A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 16, 17, 17, 18, 17, 17, 16, 15, 13, 12, 11, 10, 9, 8, 7, 7, 6, 6, 6, 6, 5, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 9, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 11, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 12, 8, 8, 8

Offset: 0

Giovanni Resta, Aug 10 2015

It is known that a(n) > 0 for every n, i.e., every number can be written as the sum of 3 palindromes.

The graph has a kind of self-similarity: looking at the first 100 values, there is a Gaussian-shaped peak centered at the first local maximum a(15) = 18. Looking at the first 10000 values, one sees just one Gaussian-shaped peak centered around the record and local maximum a(1453) = 766, but to both sides of this value there are smaller peaks, roughly at distances which are multiples of 10. In the range [1..10^6], one sees a Gaussian-shaped peak centered around the record a(164445) = 57714. In the range [1..3*10^7], there is a similar peak of height ~ 4.3*10^6 at 1.65*10^7, with smaller neighbor peaks at distances which are multiples of 10^6, etc. - M. F. Hasler, Sep 09 2018

			a(0)=1 because 0 = 0+0+0;
a(1)=1 because 1 = 0+0+1;
a(2)=2 because 2 = 0+1+1 = 0+0+2;
a(3)=3 because 3 = 1+1+1 = 0+1+2 = 0+0+3.
a(28) = 6 since 28 can be expressed in 6 ways as the sum of 3 palindromes, namely, 28 = 0+6+22 = 1+5+22 = 2+4+22 = 3+3+22 = 6+11+11 = 8+9+11.

Giovanni Resta, Table of n, a(n) for n = 0..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2016-2017.
Erich Friedman, Problem of the Month (June 1999)
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)
Hugo Pfoertner, Plot of first 10^6 terms
Hugo Pfoertner, Plot of first 3*10^7 terms
Hugo Pfoertner, Plot of low values in range 7*10^6 ... 10^7

Cf. A002113, A035137, A260254, A261131, A319453.

See A261422 for another version.

A261132 := proc(n)
    local xi,yi,x,y,z,a ;
    a := 0 ;
    for xi from 1 do
        x := A002113(xi) ;
        if 3*x > n then
            return a;
        end if;
        for yi from xi do
            y := A002113(yi) ;
            if x+2*y > n then
                break;
            else
                z := n-x-y ;
                if z >= y and isA002113(z) then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    return a;
end proc:
seq(A261132(n),n=0..80) ; # R. J. Mathar, Sep 09 2015

pal=Select[Range[0, 1000], (d = IntegerDigits@ #; d == Reverse@ d)&]; a[n_] := Length@ IntegerPartitions[n, {3}, pal]; a /@ Range[0, 1000]

A261132(n)=n||return(1); my(c=0, i=inv_A002113(n)); A2113[i] > n && i--; until( A2113[i--]*3 < n, j = inv_A002113(D = n-A2113[i]); if( j>i, j=i, A2113[j] > D && j--); while( j >= k = inv_A002113(D - A2113[j]), A2113[k] == D - A2113[j] && c++; j--||break));c \\ For efficiency, this uses an array A2113 precomputed at least up to n. - M. F. Hasler, Sep 10 2018

a(n) = Sum_{k=0..3} A319453(n,k). - Alois P. Heinz, Sep 19 2018

Examples revised and plots for large n added by Hugo Pfoertner, Aug 11 2015

A260255 Numbers that can be written as the sum of two nonnegative palindromes in base 10.

Original entry on oeis.org

0, 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, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Reinhard Zumkeller, Jul 21 2015

More than the usual number of terms are shown in order to distinguish this from A261906. - N. J. A. Sloane, Sep 09 2015

A260254(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Fraction of first n numbers that are terms of this sequence, for n < 10^8, Sep 15 2015

Cf. A035137 (complement), A260254, A002113.

111 is a member of this sequence but not of A261906. A213879 lists the differences.

a260255 n = a260255_list !! (n-1)
a260255_list = filter ((> 0) . a260254) [0..]

palQ[n_Integer, base_Integer] := Block[{}, Reverse[idn = IntegerDigits[n, base]] == idn]; Take[ Union[ Plus @@@ Tuples[ Select[ Range[0, 100], palQ[#, 10] &], 2]], 90] (* Robert G. Wilson v, Jul 22 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

A261424 Difference between n and the largest palindrome <= n.

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

Offset: 0

N. J. A. Sloane, Aug 28 2015

Up to a(301), this is the same as the sequence b(n) = least palindrome to be subtracted from n such that the difference is again a palindrome, or 10 if no such palindrome exists. But a(302) = 10 (= 302 - 292), while b(302) = 111 is the smallest palindrome P such that 302 - P is again a palindrome, 302 - 111 = 191. Similarly, b(403) = ... = b(908) = 111. For n = 1011, 1012, ..., 1110 one has a(n) = n - 1001 = 10, 11, 12, ..., 109 while b(n) = 22, 11, 44, 55, ..., 99, b(1019) = 121, b(1020) = 101, b(1021) = 22, 33, ..., 99, b(1029) = 131, 101, 10, 33, 44, ... and so on. - M. F. Hasler, Sep 08 2015

A further sequence which starts with the same values is c(n) = n-p, where p is the largest palindrome <= n such that n-p is the sum of m-1 palindromes, where m = A261675(n) is the minimal number of palindromes that add up to n. This means that c(n) = 0 (= a(n) = b(n)) if n is a palindrome; if n is the sum of 2 palindromes, then c(n) = b(n) is the smallest palindrome such that n - c(n) is again a palindrome; if n is the sum of three palindromes, then c(n) is the smallest possible sum of two palindromes such that n - c(n) is the largest possible palindrome. The numbers with A261675(n) = 3 are listed in A035137. Here, n = 1099 is the first index for which c(n) = 100 (= 99 + 1 and 1099 - 100 = 999) differs from a(n) = n - 1001 = 98 and from b(n) = 10. - M. F. Hasler, Sep 11 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A002113, A261423.

# P has list of palindromes
palfloor:=proc(n) global P; local i;
for i from 1 to nops(P) do
   if P[i]=n then return(n); fi;
   if P[i]>n then return(P[i-1]); fi;
od:
end;
[seq(n-palfloor(n),n=0..200)];

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Table[k = n;
While[Nand[palQ@ k, k > -1], k--]; n - k, {n, 0, 86}] (* Michael De Vlieger, Sep 09 2015 *)

a(n) = n - A261423(n). - M. F. Hasler, Sep 11 2015

A104444 Not the difference of two palindromes (where 0 is considered a palindrome).

Original entry on oeis.org

1020, 1029, 1031, 1038, 1041, 1047, 1051, 1061, 1065, 1071, 1074, 1081, 1091, 1101, 1130, 1131, 1139, 1141, 1148, 1151, 1157, 1161, 1171, 1175, 1181, 1191, 1201, 1231, 1240, 1241, 1249, 1251, 1258, 1261, 1267, 1271, 1281, 1291, 1301, 1314, 1341, 1350

Offset: 1

David W. Wilson, Mar 07 2005

David W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number

Cf. A002113, A035137, A084843.

A084843 \ A002113 (conjecture). [R. J. Mathar, Jul 23 2009]

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

A319477 Nonnegative integers which cannot be obtained by adding exactly two nonzero decimal palindromes.

Original entry on oeis.org

0, 1, 21, 32, 43, 54, 65, 76, 87, 98, 111, 131, 141, 151, 161, 171, 181, 191, 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

Offset: 1

Alois P. Heinz, Sep 19 2018

Every integer larger than two can be obtained by adding exactly three nonzero decimal palindromes.

The nonzero palindromes of this sequence are in A213879.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2017, Math. Comp. 87 (2018), 3023-3055.
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)

Cf. A002113, A035137 (allowing zero), A213879, A261131, A319453, A319468, A319586.

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, t) option remember; `if`(n=0, 1, `if`(t*i (k-> b(n, h(n), k)-b(n, h(n), k-1))(2):
a:= proc(n) option remember; local j; for j from 1+
      `if`(n=1, -1, a(n-1)) while g(j)<>0 do od; j
    end:
seq(a(n), n=1..80);

A319468(a(n)) = 0.

A261910 Numbers n which are neither palindromes nor the sum of two palindromes, with property that subtracting the largest palindrome < n from n gives a number which is the sum of two palindromes.

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, 1101, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1134, 1135, 1136, 1137, 1138, 1139, 1145, 1146, 1147, 1148, 1149, 1153

Offset: 1

N. J. A. Sloane, Sep 10 2015

These are the numbers with palindromic order 3 (see A261913).

More than the usual number of terms are shown in order to clarify the difference between this sequence and A035137.

N. J. A. Sloane, Table of n, a(n) for n = 1..9937

A subset of A035137.

Cf. A002113, A261675, A261911, A261912, A261913.

cp's OEIS Frontend

A261908 Numbers of the form |m - R(m)| for m in A035137, where R(m) denotes the digit-reversal of m.

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A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w.

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A260255 Numbers that can be written as the sum of two nonnegative palindromes in base 10.

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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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A261424 Difference between n and the largest palindrome <= n.

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A104444 Not the difference of two palindromes (where 0 is considered a palindrome).

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

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