A213879 -id:A213879 - cp's OEIS frontend

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

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 *)

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));

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.

A319586 Number of n-digit base-10 palindromes (A002113) that cannot be written as the sum of two positive base-10 palindromes.

Original entry on oeis.org

2, 0, 8, 7, 95, 94, 975, 971, 9810, 9805, 98288, 98272

Offset: 1

Hugo Pfoertner, Sep 23 2018

			a(1) = 2, because 0 and 1 are not sums of two positive 1-digit integers, all of which are palindromes. a(3) = 8, because the 8 3-digit palindromes 111, 131, 141, 151, 161, 171, 181, and 191 (A213879(2) ... A213879(9)) cannot be written as sum of two nonzero palindromes.

Cf. A002113, A035137, A213879, A319477.

\\ calculates a(2)...a(8) using M. F. Hasler's functions in A002113
A002113(n)={my(L=logint(n,10));(n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(nA002113(n)={Vecrev(n=digits(n))==n}
inv_A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P}
for(i=1,8,j=0;for(m=inv_A002113(10^i+1),inv_A002113(2*(10^i+1)),P=A002113(m);issum=0;for(k=2,m,PP=A002113(k);if(PP>P/2,break);if(is_A002113(P-PP),issum=1;break));if(issum==0,j++));print1(j,", ",))

from sympy import isprime
from itertools import product
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def a(n):
    palslst = [p for d in range(1, n+1) for p in pals(d)][1:]
    palsset = set(palslst)
    cs = ctot = 0
    for p in pals(n):
        ctot += 1
        for p1 in palslst:
            if p - p1 in palsset: cs += 1; break
            if p1 > p//2: break
    return ctot - cs
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Jul 12 2021

a(12) from Giovanni Resta, Oct 01 2018

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

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

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A319586 Number of n-digit base-10 palindromes (A002113) that cannot be written as the sum of two positive base-10 palindromes.

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