A356824 -id:A356824 - cp's OEIS frontend

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

Tanya Khovanova and Massimo Kofler, Aug 31 2022

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.

			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.

Cf. A356824.

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]]]

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 pMichael S. Branicky, Aug 31 2022

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

Original entry on oeis.org

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

Tanya Khovanova, Sep 02 2022

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.

			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.

Cf. A356824, A356854.

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]]]

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+qMichael S. Branicky, Sep 02 2022

cp's OEIS Frontend

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

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