cp's OEIS Frontend

A371074 Number of the rightmost decimal digits of n that are the same as those of n^n.

%I A371074 #18 Mar 30 2024 11:32:58
%S A371074 0,1,0,0,0,1,1,0,0,1,1,2,0,1,0,1,2,1,0,1,1,2,0,0,0,2,1,0,0,1,1,2,0,1,
%T A371074 0,1,2,1,0,1,1,2,0,0,0,1,1,0,0,2,1,3,0,1,0,1,2,3,0,1,1,2,0,0,0,1,1,0,
%U A371074 0,1,1,2,0,1,0,2,2,1,0,1,1,2,0,0,0,1,1
%N A371074 Number of the rightmost decimal digits of n that are the same as those of n^n.
%C A371074 The common digits might include leading 0's (such as at n = 51 or n = 57) and they are included in the total.
%C A371074 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 which is all the rightmost 0's of n.
%C A371074 For every n >= 0, a(n) is the congruence speed of n at height 1 by Definitions 1.1 and 1.3 of the paper entitled "Number of stable digits of any integer tetration" (see Links).
%H A371074 Jorge Jiménez Urroz and José Luis Andrés Yebra, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Yebra/yebra4.html">On the Equation a^x == x (mod b^n)</a>, Journal of Integer Sequences, Article 09.8.8, 2009.
%H A371074 Marco Ripà, <a href="https://arxiv.org/abs/2402.07929">Congruence speed of tetration bases ending with 0</a>, arXiv:2402.07929 [math.NT], 2024.
%H A371074 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 A371074 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A371074 For any n >= 2, a(n) is such that n == n^n (mod 10^(a(n))) and n <> n^n (mod 10^(a(n)+1)).
%e A371074 a(0) = 0 since 0^0 = 1 so that 0 and 0^0 have no digits in common.
%e A371074 For n = 51, a(n) = 3 since 51^51 == 5051 (mod 10^4).
%Y A371074 Cf. A000312, A082576, A317905, A349425, A369826, A369624, A370211.
%K A371074 nonn,base
%O A371074 0,12
%A A371074 _Marco Ripà_, Mar 10 2024