A319586 Number of n-digit base-10 palindromes (A002113) that cannot be written as the sum of two positive base-10 palindromes.

2, 0, 8, 7, 95, 94, 975, 971, 9810, 9805, 98288, 98272

Offset

1,1

Links

Table of n, a(n) for n=1..12.

Example

a(1) = 2, because 0 and 1 are not sums of two positive 1-digit integers, all of which are palindromes. a(3) = 8, because the 8 3-digit palindromes 111, 131, 141, 151, 161, 171, 181, and 191 (A213879(2) ... A213879(9)) cannot be written as sum of two nonzero palindromes.

Prog

(PARI) \\ calculates a(2)...a(8) using M. F. Hasler's functions in A002113
A002113(n)={my(L=logint(n, 10)); (n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(n<L, n, n\10))))}
is_A002113(n)={Vecrev(n=digits(n))==n}
inv_A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P}
for(i=1, 8, j=0; for(m=inv_A002113(10^i+1), inv_A002113(2*(10^i+1)), P=A002113(m); issum=0; for(k=2, m, PP=A002113(k); if(PP>P/2, break); if(is_A002113(P-PP), issum=1; break)); if(issum==0, j++)); print1(j, ", ", ))
(Python)
from sympy import isprime
from itertools import product
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def a(n):
    palslst = [p for d in range(1, n+1) for p in pals(d)][1:]
    palsset = set(palslst)
    cs = ctot = 0
    for p in pals(n):
        ctot += 1
        for p1 in palslst:
            if p - p1 in palsset: cs += 1; break
            if p1 > p//2: break
    return ctot - cs
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Jul 12 2021

Crossrefs

Cf. A002113, A035137, A213879, A319477.

Sequence in context: A366164 A021483 A011015 * A287467 A288015 A288438

Adjacent sequences: A319583 A319584 A319585 * A319587 A319588 A319589

Keyword

nonn,base,hard,more

Author

Hugo Pfoertner, Sep 23 2018

Extensions

a(12) from Giovanni Resta, Oct 01 2018

Status

approved