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A350588 a(n) is the number of distinct numbers of steps required for the last n digits of integers to repeat themselves by iterating the map m -> m^5.

%I A350588 #5 Feb 17 2022 01:11:13
%S A350588 1,2,3,4,6,9,14,23,33,45,59,75,93,113,135,159,184,211,240,271,304,339,
%T A350588 376,415,456,499,544,591,640,691,744,799,855,913,973,1035,1099,1165,
%U A350588 1233,1303,1375,1449,1525,1603,1683,1765,1849,1935,2023,2113,2205,2299
%N A350588 a(n) is the number of distinct numbers of steps required for the last n digits of integers to repeat themselves by iterating the map m -> m^5.
%F A350588 For n >= 9, a(n) = a(n-1) + 2*n - 4 - ceiling(log_2 (n)) or a(n) = n^2 - 3*n - 17 - Sum_{i=9..n} ceiling(log_2 (i)).
%e A350588 a(1) = 1. It takes one step to repeat the last digit by iterating the map on an integer. For example, 2^5 = 32 and 9^5 = 59049. Thus, the distinct number of steps for n = 1 is {1} and a(1) = 1.
%e A350588 a(2) = 2. It takes 1 or 2 steps for an integer to repeat its last two digits. For example, 24 -> 7962624; 27 -> 14348907 -> 608266787713357709119683992618861307. Thus, a(2) = 2: {1, 2}.
%e A350588 a(3)  =  3: {1..3}.
%e A350588 a(4)  =  4: {1..4}.
%e A350588 a(5)  =  6: {1..6}.
%e A350588 a(6)  =  9: {1..9}.
%e A350588 a(7)  = 14: {1..14}.
%e A350588 a(8)  = 23: {1..23}.
%e A350588 a(9)  = 33: {1..24, 32..40}.
%e A350588 a(10) = 45: {1..25, 32..41, 64..73}.
%e A350588 a(11) = 59: {1..26, 32..42, 64..74, 128..138}.
%o A350588 (Python)
%o A350588 from math import log, ceil
%o A350588 def A350588(n):
%o A350588     if n <= 8:
%o A350588         b, S = 10**n, set()
%o A350588         for i in range(b):
%o A350588             t, s, T = i, 0, set()
%o A350588             while t not in T: T.add(t); t = (t**5)%b; s += 1
%o A350588             S.add(s)
%o A350588         return(len(S))
%o A350588     else: return n*n - 3*n - 17 - sum(ceil(log(i, 2)) for i in range(9, n+1))
%Y A350588 Cf. A000584, A348338, A348339, A349744.
%K A350588 nonn,base
%O A350588 1,2
%A A350588 _Ya-Ping Lu_, Jan 07 2022