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

Author

Hugo Pfoertner, Sep 23 2018

Keywords

• nonn
• base
• hard
• more

Examples

			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.
		

Crossrefs

Cf. A002113, A035137, A213879, A319477.

Programs

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(nA002113(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

Extensions

a(12) from Giovanni Resta, Oct 01 2018