A308803 a(n) is the largest n-digit palindrome that is the product of two numbers having an equal number of digits, or 0 if no such palindrome exists.

9, 0, 999, 9009, 99999, 906609, 9999999, 99000099, 999969999, 9966006699, 99999999999, 999000000999, 9999998999999, 99956644665999, 999999999999999, 9999000000009999, 99999999799999999, 999900665566009999, 9999999997999999999, 99999834000043899999, 999999999999999999999, 9999994020000204999999, 99999999999899999999999

Offset: 1

Author

Donghwi Park, May 06 2019

Keywords

• nonn
• base

Comments

From Chai Wah Wu, Sep 30 2019: (Start)

Note that the product decomposition satisfying the conditions is not necessarily unique. For instance, a(5) = 99999 = 369*271 = 123*813 and a(9) = 50001*19999 = 16667*59997.

When n is odd, a(n) in decimal are all 9's with the possible exception of the middle digit which can be 6,7,8 or 9, i.e. a(n) = 10^n-1-k*10^((n-1)/2) for some 0 <= k <= 3.

In particular, a(2m+1) >= (2*10^m-1)(5*10^m+1) = 10^(2m+1)-3*10^m-1. This inequality is an equality for m = 4, 15, 18, 20, 23, 29, 33, 34, 35.

See the "Decomposition of a(n) for odd n" file in the Links section for examples.

a(4m) >= (10^(2m)-1)(10^(2m)-10^m+1). The inequality is strict for m = 5. Is this a rare occurrence?

(End)

a(24) = 999999000001 * 999999999999 = 999999000000000000999999. - David A. Corneth, Sep 30 2019

Examples

			a(1)=9 because 3*3=9;
a(2)=0 because there is no such palindrome;
a(3)=999 because 27*37=999;
a(4)=9009 because 99*91=9009;
a(5)=99999 because 369*271=99999;
a(6)=906609 because 993*913=906609;
a(7)=9999999 because 2151*4649=9999999;
a(8)=99000099 because 9999*9901=99000099;
a(9)=999969999 because 50001*19999=999969999;
a(10)=9966006699 because 99979*99681=9966006699;
a(11)=99999999999 because 194841*513239=99999999999;
a(12)=999000000999 because 999999*999001=999000000999;
a(13)=9999998999999 because 2893921*3455519=9999998999999.
		

Links

• Math StackExchange, Proof for all largest palindrome numbers for each power of 10
• Donghwi Park, Source code for even n and all examples for n<=21
• Chai Wah Wu, Decomposition of a(n) for odd n

Crossrefs

Cf. A327897. Subsequence of A002113.

Extensions

a(14)-a(20) from Jon E. Schoenfield, May 10 2019

a(21) from Donghwi Park, Jul 16 2019

a(22)-a(23) from Chai Wah Wu, Sep 30 2019

a(20) corrected by Donghwi Park, Dec 18 2020