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

1,1

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

Links

Table of n, a(n) for n=1..23.

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

Example

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.

Crossrefs

Cf. A327897. Subsequence of A002113.

Sequence in context: A178912 A191564 A067479 * A249489 A166734 A155783

Adjacent sequences: A308800 A308801 A308802 * A308804 A308805 A308806

Keyword

nonn,base

Author

Donghwi Park, May 06 2019

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

Status

approved