A261906 -id:A261906 - cp's OEIS frontend

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

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

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 *)

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));

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.

A356824 Palindromes that can be written as the sum of two palindromic primes.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 22, 202, 232, 252, 262, 282, 292, 414, 444, 454, 464, 474, 484, 494, 626, 666, 686, 696, 808, 828, 858, 878, 888, 898, 20002, 20602, 20802, 20902, 21612, 21712, 21812, 21912, 22622, 22722, 22822, 22922, 23632, 23732, 23832, 23932, 24642, 24742, 24842, 24942

Offset: 1

Tanya Khovanova, Aug 29 2022

With the exception of 22, which is the sum of 11 and 11, no term of this sequence has an even number of digits. Proof: Other than 11, palindromes with an even number of digits are not primes (since they are divisible by 11). Suppose m is a term of this sequence with 2k digits. Then m must be the sum of two palindromic primes p and q with 2k-1 digits each. It follows that the first and the last digit of m is 1. Hence, either p or q is even, creating a contradiction with primality.

With the exception of 5, 7, and 9, all terms of this sequence are even. Proof: two consecutive multi-digit palindromes differ by at least 10, so larger palindromes can't be the sum of a palindromic prime and 2. Thus, each multi-digit term is the sum of two odd numbers.

			282 can be written as the sum of two prime palindromes, 101 and 181. Thus, 282 is in the sequence.

Cf. A002113, A002385, A261906.

q := Select[Range[30000], PalindromeQ[#] && PrimeQ[#] &]
Select[Union[Flatten[Table[q[[n]] + q[[m]], {n, Length[q]}, {m, Length[q]}]]],
PalindromeQ[#] &]

from sympy import isprime
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def oddpals(d): # generator of odd palindromes with d digits
    if d == 1: yield from [1, 3, 5, 7, 9]; return
    for first in "13579":
        for p in product("0123456789", repeat=(d-2)//2):
            left = "".join(p); right = left[::-1]
            for mid in [[""], "0123456789"][d%2]:
                yield int(first + left + mid + right + first)
def auptod(dd):
    N, alst, pp = 10**dd, [], [2, 3, 5, 7, 11]
    pp += [p for d in range(3, dd+1, 2) for p in oddpals(d) if isprime(p)]
    return sorted(set(p+q for p in pp for q in pp if p+qMichael S. Branicky, Aug 29 2022

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}]]

A262173 Number of nonnegative integers < 10^n that are palindromes or the sum of 2 palindromes.

Original entry on oeis.org

1, 10, 92, 991, 8012, 90970, 733052, 8861377, 68729295, 875790193

Offset: 0

Chai Wah Wu, Sep 18 2015

Cf. A260255, A261906, A261907.

A309325 Numbers that are the sum of two successive palindromes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 20, 33, 55, 77, 99, 121, 143, 165, 187, 200, 212, 232, 252, 272, 292, 312, 332, 352, 372, 393, 414, 434, 454, 474, 494, 514, 534, 554, 574, 595, 616, 636, 656, 676, 696, 716, 736, 756, 776, 797, 818, 838, 858, 878, 898, 918, 938, 958, 978

Offset: 1

Ilya Gutkovskiy, Jul 23 2019

Index entries for sequences related to palindromes

Cf. A002113, A046489, A046497, A260255, A261906.

Total /@ Partition[Select[Range[0, 500], PalindromeQ], 2, 1]

from itertools import chain, count, islice
def A309325_gen(): # generator of terms
    c = 0
    for a in chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l,10**(l+1)))) for l in count(0)):
        yield c+(c:=a)
A309325_list = list(islice(A309325_gen(),20)) # Chai Wah Wu, Jun 23 2022

a(n) = A002113(n) + A002113(n+1).

cp's OEIS Frontend

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

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

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

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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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A356824 Palindromes that can be written as the sum of two palindromic primes.

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

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A262173 Number of nonnegative integers < 10^n that are palindromes or the sum of 2 palindromes.

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A309325 Numbers that are the sum of two successive palindromes.

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