A256162 Positive integers a(n) such that number of digits in decimal expansion of a(n)^a(n) is divisible by a(n).

1, 8, 9, 98, 99, 998, 999, 9998, 9999, 99998, 99999, 999998, 999999, 9999998, 9999999, 99999998, 99999999, 999999998, 999999999, 9999999998, 9999999999, 99999999998, 99999999999, 999999999998, 999999999999, 9999999999998, 9999999999999

Offset: 1

Author

Bui Quang Tuan, Mar 17 2015

Keywords

• nonn
• base

Comments

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

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

Examples

			1^1 = 1 has 1 digit, and 1 is divisible by 1.
8^8 = 16777216 has 8 digits, and 8 is divisible by 8.
98^98 has 196 digits, and 196 is divisible by 98.
		

Links

• Bui Quang Tuan, Table of n, a(n) for n = 1..101

Crossrefs

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

Programs

Magma

[1] cat [10^Floor((n+1)/2)-2*Floor((n+1)/2)+n-1: n in [1..30]]; // Vincenzo Librandi, Mar 18 2015

Mathematica

Select[Range@10000, Mod[IntegerLength[#^#], #] == 0 &] (* Michael De Vlieger, Mar 17 2015 *)
Join[{1}, Table[(10^Floor[n/2] - 2 Floor[n/2] + n - 2), {n, 2, 30}]] (* Vincenzo Librandi, Mar 18 2015 *)

PARI

isok(n) = !(#digits(n^n) % n); \\ Michel Marcus, Mar 17 2015

Formula

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.