Demlo Number

The initially palindromic numbers 1, 121, 12321, 1234321, 123454321, ... (Sloane's A002477). For the first through ninth terms, the sequence is given by the generating function

(1)

(Plouffe 1992, Sloane and Plouffe 1995).

The definition of this sequence is slightly ambiguous from the tenth term on, but the most common convention follows from the following observation. The sequences of consecutive and reverse digits  and , respectively, are given by

			(2)
			(3)

for , so the first few Demlo numbers are given by

			(4)
			(5)

But, amazingly, this is just the square of the th repunit , i.e.,

(6)

for , and the squares of the first few repunits are precisely the Demlo numbers: , , , ... (Sloane's A002275 and A002477). It is therefore natural to use (6) as the definition for Demlo numbers  with , giving 1, 121, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ....

The equality  for  also follows immediately from schoolbook multiplication, as illustrated above. This follows from the algebraic identity

(7)

The sums of digits of the Demlo numbers for  are given by

(8)

More generally, for , 2, ..., the sums of digits are 1, 4, 9, 16, 25, 36, 49, 64, 81, 82, 85, 90, 97, 106, ... (Sloane's A080151). The values of  for which these are square are 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 51, 66, 81, ... (Sloane's A080161), corresponding to the Demlo numbers 1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345679012345679012345679012345678987654320987654320987654320987654321, ... (Sloane's A080162).

Ref: mathworld.wolfram.com

Palindromic Number

A palindromic number is a number (in some base ) that is the same when written forwards or backwards, i.e., of the form . The first few palindromic numbers are therefore are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ... (Sloane's A002113). The number of palindromic numbers less than a given number are illustrated in the plot above.

The numbers of palindromic numbers less than 10, , , ... are 9, 18, 108, 198, 1098, 1998, 10998, ... (Sloane's A050250). This sequence is given by the closed-form formula

(1)

Banks et al. (2004) proved that almost all palindromes (in any base) are composite, with the precise statement being

(2)

where  is the number of palindromic primes  and  is the number of palindromic numbers .

The sum of the reciprocals of the palindromic numbers converges to a constant  (Sloane's A118031; Rivera), where the value has been computed using all palindromic numbers  is 3.370001832....

The first few  for which the pronic number  is palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane's A028336), and the first few palindromic numbers which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane's A028337). The first few numbers whose squares are palindromic are 1, 2, 3, 11, 22, 26, ... (Sloane's A002778), and the first few palindromic squares are 1, 4, 9, 121, 484, 676, ... (Sloane's A002779).

There are no palindromic square -digit numbers for , 4, 8, 10, 14, 18, 20, 24, 30, ... (Sloane's A034822).

Numbers that are not the sum of two palindromes (where 0 is itself considered a palindrome) are 21, 32, 43, 54, 65, 76, 87, 98, 201, 1031, ... (Sloane's A035137). Numbers that are not the difference of two palindromes are 1020, 1029, 1031, 1038, 1041, 1047, 1051, 1061, ... (Sloane's A104444).

Ref: mathworld.wolfram.com