A327897 a(n) is the largest palindromic number formed from two numbers with n digits multiplied together.

9, 9009, 906609, 99000099, 9966006699, 999000000999, 99956644665999, 9999000000009999, 999900665566009999, 99999834000043899999, 9999994020000204999999, 999999000000000000999999, 99999963342000024336999999, 9999999000000000000009999999, 999999974180040040081479999999

Offset: 1

Author

Christopher Shaw, Sep 29 2019

Keywords

• nonn
• base

Comments

No formula is known to the author.

Examples

			a(2) = 99 * 91 = 9009, a(3) = 993 * 913 = 906609.
		

Crossrefs

Cf. A308803.

Programs

Python

def is_palindrome(n):
    if n<10: return True
    n = str(n)
    midpoint = int(len(n)/2)
    return n[:midpoint] == n[-midpoint:][::-1]
def A327897(n):
    lower_bound = 10**(n-1) - 1
    upper_bound = 10**n - 1
    max_palindromes = (0,0,0)
    for n1 in range(upper_bound, lower_bound, -1):
        for n2 in range(n1, lower_bound, -1):
            n = n1* n2
            if is_palindrome(n) and n>max_palindromes[2]:
                max_palindromes = (n1, n2, n)
            if n < max_palindromes[2]:
                break
        if n1*n1 < max_palindromes[2]:
            break
    return max_palindromes
if _name_ == '_main_':
    for n in range(1,7):
        print(A327897(n))

Formula

a(n) = A308803(2*n) for n > 1. - Andrew Howroyd, Sep 30 2019

a(2n) >= (10^(2n)-1)*(10^(2n)-10^n+1). - Chai Wah Wu, Sep 30 2019

Extensions

a(11) from Chai Wah Wu, Sep 30 2019

a(12) from David A. Corneth, Sep 30 2019

a(13)-a(15) from Giovanni Resta, Oct 04 2019