A222809 Number of n-digit numbers N such that the reversal of N divides N but is different from N.

0, 9, 21, 122, 228, 1167, 2123, 11270, 20440, 110971, 201475, 1103592, 2005388

Offset: 1

Author

N. J. A. Sloane, Mar 10 2013

Keywords

• nonn
• base
• more

Comments

Suggested by A214927.

Conjecture: a(n) = A222811(n) - 9*10^floor((n-1)/2). - Lars Blomberg, Jul 03 2014

Proof of this conjecture: A number N with n digits which equals its own reversal has the first and identical last digit from {1, 2, ..., 9}. If n is even the other n-2 digits come in n/2 - 1 pairs of equal numbers from {0, 1, ..., 9}. If n is odd with n >= 3 then the other n-2 numbers come in (n-3)/2 pairs of equal numbers from {0, 1, ..., 9} and an additional middle digit also from {0, 1, ..., 9}. Therefore there are 9*10^(n/2-1) such numbers N for even n, and 9*10^((n-1)/2) for odd n, fitting 9*10^floor((n-1)/2). - Wolfdieter Lang, Jul 13 2014

Examples

			Some of the smallest solutions are:
[10, 20, 30, 40, 50, 60, 70, 80, 90] (so a(2) = 9),
[100, 110, 200, 220, 300, 330, 400, 440, 500, 510, 540, 550, 600, 660, 700, 770, 800, 810, 880, 900, 990] (so a(3) = 21),
[1000, 1010, 1100, 1110, 1210, 1310, 1410, ...].
		

Crossrefs

Cf. A214927, A222810, A222811, A222812.

Programs

PARI

a(n) = sum(i=10^(n-1), 10^n-1, (irev=eval(concat(Vecrev(Str(i))))) && irev!=i && !(i % irev)); \\ Michel Marcus, Jul 03 2014

Extensions

a(7)-a(12) from Lars Blomberg, Jul 03 2014

a(13) from Giovanni Resta, Aug 15 2019