A369771 - OEIS

%I #23 Apr 02 2024 21:45:55

%S 1,0,1,1,2,2,8,4,4,2,3,10000000000,4,2,3,2,13,4,3,2,3,

%T 104857600000000000000000000,4,1,2,4,12,8,2,2,3,

%U 205891132094649000000000000000000000000000000,4,4,3,2,7,4,3,1,3,12089258196146291747061760000000000000000000000000000000000000000

%N Number of the rightmost decimal digits of n^(n^n) that are the same as those of n^(n^(n^n)).

%C The common digits might include leading 0's (such as at n = 5) and they are included in the total.

%C Let c be a positive integer and assume that k is a positive integer that is not a multiple of 10. If n = k*10^c, then a(n) = c*(n^n) which is all the rightmost 0's of n^(n^n).

%H Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.

%F For any n >=2, a(n) is such that n^(n^n) == n^(n^(n^n)) (mod 10^(a(n))) and n^(n^n) <> n^(n^(n^n)) (mod 10^(a(n)+1)).

%e a(-1) = 1 since (-1)^(-1) = -1 (which is a one-digit number);

%e a(0) = 0 since 0^0 = 1 so that 0^(0^0) = 0 and 0^(0^(0^0)) = 1 have no digits in common.

%e For n=5, a(n)=8 since 5^(5^5) == 908203125 (mod 10^9) while 5^(5^(5^5)) == 408203125 (mod 10^9).

%Y Cf. A002488, A002489, A317905, A349425, A369624.

%K sign,base

%O -1,5

%A _Marco Ripà_, Jan 31 2024