Define a(N) as the smallest integer k such that the
digits of N are at the Nth place in N^k
(0 if there is no such k)
Example : a(13) = 17, because 13^17 = 8650415919381337933
, and starting at the 13th place, we have 13.
The first terms of this sequence are
1 1
2 5
3 5
4 5
5 6
6 7
7 32
8 10
9 9
10 0
11 11
12 81
13 17
14 64
15 74
16 347
Directed graph of the 1000
first terms (a(x)=y -> a(y)=z ->
a(z)=t etc.)
N such that the first occurrence of the digits of N is at the Ntn place
in N^N :
1,9,11,276,708,9118,61051,77527
(no other term up to 100000)
Statistically, the number of terms up to y = 10^x – 1 should be 1 + 0.1*8+
0.9*(x – 1)
So, up to 99999 there should be about 5.4 terms (in fact there are 8, I
guess we're lucky here :o)
Jean-Marc Falcoz
April 2009
