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A356854
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Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.
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1
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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
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listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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This sequence doesn't contain any numbers with an even number of digits, see proof in 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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]]]
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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