A356854 Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.

282, 484, 858, 888, 21912, 22722, 23832, 24642, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28782, 28882, 28982, 29692, 29792, 29892, 29992, 40704, 41514, 41614, 41814, 42624, 42824, 42924, 43434, 43734, 43834, 43934, 44744, 44844, 44944, 45354

Offset

1,1

Comments

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

Subsequence of A356824.

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..40.

Example

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.

Mathematica

q := Select[Range[50000], PalindromeQ[#] && PrimeQ[#] &]
Sort[Transpose[Select[Tally[Flatten[Table[q[[n]] + q[[m]], {n, Length[q]}, {m, n + 1, 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, Aug 31 2022

Crossrefs

Cf. A356824.

Sequence in context: A242982 A255387 A320706 * A114804 A250580 A252364

Adjacent sequences: A356851 A356852 A356853 * A356855 A356856 A356857

Keyword

nonn,base

Author

Tanya Khovanova and Massimo Kofler, Aug 31 2022

Status

approved