The OEIS is supported by the many generous donors to the OEIS Foundation.

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

1

`%I #11 Sep 04 2022 12:46:38
`

`%S 282,484,858,888,21912,22722,23832,24642,25752,26662,26762,26862,
`

`%T 26962,27672,27772,27872,27972,28482,28782,28882,28982,29692,29792,
`

`%U 29892,29992,40704,41514,41614,41814,42624,42824,42924,43434,43734,43834,43934,44744,44844,44944,45354
`

`%N Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.
`

`%C This sequence doesn't contain any numbers with an even number of digits, see proof in A356824.
`

`%C Subsequence of A356824.
`

`%C 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 282 can be expressed as a sum of two distinct palindromic primes in two ways: 282 = 101 + 181 = 131 + 151. Thus, 282 is in this sequence.
`

`%t q := Select[Range[50000], PalindromeQ[#] && PrimeQ[#] &]
`

`%t Sort[Transpose[Select[Tally[Flatten[Table[q[[n]] + q[[m]], {n, Length[q]}, {m, n + 1, Length[q]}]]], PalindromeQ[#[[1]]] && #[[2]] > 1 &]][[1]]]
`

`%o (Python)
`

`%o from sympy import isprime
`

`%o from itertools import product
`

`%o def ispal(n): s = str(n); return s == s[::-1]
`

`%o def oddpals(d): # generator of odd palindromes with d digits
`

`%o if d == 1: yield from [1, 3, 5, 7, 9]; return
`

`%o for first in "13579":
`

`%o for p in product("0123456789", repeat=(d-2)//2):
`

`%o left = "".join(p); right = left[::-1]
`

`%o for mid in [[""], "0123456789"][d%2]:
`

`%o yield int(first + left + mid + right + first)
`

`%o def auptod(dd):
`

`%o N, alst, pp, once, twice = 10**dd, [], [2, 3, 5, 7, 11], set(), set()
`

`%o pp += [p for d in range(3, dd+1, 2) for p in oddpals(d) if isprime(p)]
`

`%o sums = (p+q for p in pp for q in pp if p<q and p+q<N and ispal(p+q))
`

`%o for s in sums:
`

`%o if s in once: twice.add(s)
`

`%o else: once.add(s)
`

`%o return sorted(twice)
`

`%o print(auptod(5)) # _Michael S. Branicky_, Aug 31 2022
`

`%Y Cf. A356824.
`

`%K nonn,base
`

`%O 1,1
`

`%A _Tanya Khovanova_ and _Massimo Kofler_, Aug 31 2022
`