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A356881 Palindromes that can be written in more than one way as the sum of two palindromic primes.

%I A356881 #11 Feb 22 2024 20:11:30
%S A356881 202,282,484,858,888,21912,22722,23832,24642,24842,25752,26662,26762,
%T A356881 26862,26962,27672,27772,27872,27972,28482,28682,28782,28882,28982,
%U A356881 29692,29792,29892,29992,40704,41514,41614,41814,42624,42824,42924,43434,43734
%N A356881 Palindromes that can be written in more than one way as the sum of two palindromic primes.
%C A356881 This sequence doesn't contain any numbers with an even number of digits, see proof in A356824.
%C A356881 Subsequence of A356824.
%C A356881 Supersequence of A356854, which requires the two palindromic primes to be distinct. For example, 202, 24842, and 28682 are in this sequence but not in A356854.
%C A356881 All numbers in this sequence are even. Proof: any 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 term is the sum of two odd numbers.
%e A356881 282 can be expressed as the sum of two palindromic primes in two ways: 282 = 101 + 181 = 131 + 151. Thus, 282 is in this sequence. Similarly, 202 = 101 + 101 = 11 + 191.
%t A356881 q := Select[Range[50000], PalindromeQ[#] && PrimeQ[#] &]Sort[Transpose[Select[Tally[ Flatten[Table[ q[[n]] + q[[m]], {n, Length[q]}, {m, n, Length[q]}]]],PalindromeQ[#[[1]]] && #[[2]] > 1 &]][[1]]]
%o A356881 (Python)
%o A356881 from sympy import isprime
%o A356881 from itertools import product
%o A356881 def ispal(n): s = str(n); return s == s[::-1]
%o A356881 def oddpals(d): # generator of odd palindromes with d digits
%o A356881     if d == 1: yield from [1, 3, 5, 7, 9]; return
%o A356881     for first in "13579":
%o A356881         for p in product("0123456789", repeat=(d-2)//2):
%o A356881             left = "".join(p); right = left[::-1]
%o A356881             for mid in [[""], "0123456789"][d%2]:
%o A356881                 yield int(first + left + mid + right + first)
%o A356881 def auptod(dd):
%o A356881     N, alst, pp, once, twice = 10**dd, [], [2, 3, 5, 7, 11], set(), set()
%o A356881     pp += [p for d in range(3, dd+1, 2) for p in oddpals(d) if isprime(p)]
%o A356881     sums = (p+q for p in pp for q in pp if p<=q and p+q<N and ispal(p+q))
%o A356881     for s in sums:
%o A356881         if s in once: twice.add(s)
%o A356881         else: once.add(s)
%o A356881     return sorted(twice)
%o A356881 print(auptod(5)) # _Michael S. Branicky_, Sep 02 2022
%Y A356881 Cf. A356824, A356854.
%K A356881 nonn,base
%O A356881 1,1
%A A356881 _Tanya Khovanova_, Sep 02 2022