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

202, 282, 484, 858, 888, 21912, 22722, 23832, 24642, 24842, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28682, 28782, 28882, 28982, 29692, 29792, 29892, 29992, 40704, 41514, 41614, 41814, 42624, 42824, 42924, 43434, 43734

Offset

1,1

Comments

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

Subsequence of A356824.

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.

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.

Links

Table of n, a(n) for n=1..37.

Example

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.

Mathematica

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]]]

Prog

(Python)
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, once, twice = 10**dd, [], [2, 3, 5, 7, 11], set(), set()
    pp += [p for d in range(3, dd+1, 2) for p in oddpals(d) if isprime(p)]
    sums = (p+q for p in pp for q in pp if p<=q and p+q<N and ispal(p+q))
    for s in sums:
        if s in once: twice.add(s)
        else: once.add(s)
    return sorted(twice)
print(auptod(5)) # Michael S. Branicky, Sep 02 2022

Crossrefs

Cf. A356824, A356854.

Sequence in context: A166505 A066130 A067335 * A235082 A260280 A252993

Adjacent sequences: A356878 A356879 A356880 * A356882 A356883 A356884

Keyword

nonn,base

Author

Tanya Khovanova, Sep 02 2022

Status

approved