%I #35 Sep 08 2022 08:46:11

%S 1,8,9,98,99,998,999,9998,9999,99998,99999,999998,999999,9999998,

%T 9999999,99999998,99999999,999999998,999999999,9999999998,9999999999,

%U 99999999998,99999999999,999999999998,999999999999,9999999999998,9999999999999

%N Positive integers a(n) such that number of digits in decimal expansion of a(n)^a(n) is divisible by a(n).

%C A055642(a(n)^a(n)) = A055642(a(n))*a(n).

%C 1 + floor(log_10(a(n)^a(n))) = a(n)*(1 + floor(log_10(a(n)))).

%H Bui Quang Tuan, <a href="/A256162/b256162.txt">Table of n, a(n) for n = 1..101</a>

%F a(n) = 10^floor(n/2) - 2*floor(n/2) + n - 2 = 10^floor(n/2)-(1+(-1)^n)/2 - 1 for n>1, a(1) = 1.

%e 1^1 = 1 has 1 digit, and 1 is divisible by 1.

%e 8^8 = 16777216 has 8 digits, and 8 is divisible by 8.

%e 98^98 has 196 digits, and 196 is divisible by 98.

%t Select[Range@10000, Mod[IntegerLength[#^#], #] == 0 &] (* _Michael De Vlieger_, Mar 17 2015 *)

%t Join[{1}, Table[(10^Floor[n/2] - 2 Floor[n/2] + n - 2), {n, 2, 30}]] (* _Vincenzo Librandi_, Mar 18 2015 *)

%o (PARI) isok(n) = !(#digits(n^n) % n); \\ _Michel Marcus_, Mar 17 2015

%o (Magma) [1] cat [10^Floor((n+1)/2)-2*Floor((n+1)/2)+n-1: n in [1..30]]; // _Vincenzo Librandi_, Mar 18 2015

%Y Cf. A055642 (Number of digits in decimal expansion of n).

%K nonn,base

%O 1,2

%A _Bui Quang Tuan_, Mar 17 2015