cp's OEIS Frontend

A371048 Numbers formed by the rightmost decimal digits of n that are the same as those n^n, where -1 indicates that n <> n^n (mod 10).

%I A371048 #31 Mar 30 2024 11:31:35
%S A371048 0,1,-1,-1,-1,5,6,-1,-1,9,0,11,-1,3,-1,5,16,7,-1,9,0,21,-1,-1,-1,25,6,
%T A371048 -1,-1,9,0,31,-1,3,-1,5,36,7,-1,9,0,41,-1,-1,-1,5,6,-1,-1,49,0,51,-1,
%U A371048 3,-1,5,56,57,-1,9,0,61,-1,-1,5,6,-1,-1,9,0,71,-1,3
%N A371048 Numbers formed by the rightmost decimal digits of n that are the same as those n^n, where -1 indicates that n <> n^n (mod 10).
%C A371048 The common digits might include leading 0's (such as at n = 51 or n = 57) and they are discarded (in particular, a(0) = 0 indicates that the corresponding zero digit term results in a 0 integer entry).
%C A371048 Assuming that c > 0 is an integer not a multiple of 10, then a(c*10^k) = 0 for every positive integer k, since (c*10^k) and (c*10^k)^(c*10^k) have in common only their rightmost k digits.
%C A371048 a(n) is equal to -1 if and only if n == 2,4,8 (mod 10) or n == 3,7 (mod 20).
%C A371048 A082576 is a subsequence of the present one.
%H A371048 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 A371048 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.
%F A371048 If n <> 2,4,8 (mod 10) or n <> 3,7 (mod 20), then a(n) = n (mod 10^k), where k is such that n == n^n (mod 10^k) and n <> n^n (mod 10^(k+1)), whereas a(n) = -1 otherwise.
%e A371048 For n = 51, 51^51 = 1219211305094648479473193481872927834667576992593770717189298225284399541977208231315051 and 51^51 == 5051 (mod 10^4), so there might be three common final digits by including a leading 0 that should instead be disregarded. Consequently, a(51) = 51.
%Y A371048 Cf. A000312, A082576, A317905, A349425, A369624, A369826, A370211.
%K A371048 sign,base
%O A371048 0,6
%A A371048 _Marco Ripà_, Mar 10 2024