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Number of n-digit numbers N such that the reversal of N divides N but is different from N.
4

%I #23 Aug 15 2019 03:28:45

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

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

%C Suggested by A214927.

%C Conjecture: a(n) = A222811(n) - 9*10^floor((n-1)/2). - _Lars Blomberg_, Jul 03 2014

%C 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

%e Some of the smallest solutions are:

%e [10, 20, 30, 40, 50, 60, 70, 80, 90] (so a(2) = 9),

%e [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),

%e [1000, 1010, 1100, 1110, 1210, 1310, 1410, ...].

%o (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

%Y Cf. A214927, A222810, A222811, A222812.

%K nonn,base,more

%O 1,2

%A _N. J. A. Sloane_, Mar 10 2013

%E a(7)-a(12) from _Lars Blomberg_, Jul 03 2014

%E a(13) from _Giovanni Resta_, Aug 15 2019