This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348338 #33 Nov 19 2021 07:42:35
%S A348338 1,4,9,15,23,33,45,59,75,93,113,135,159,185,213,243,274,307,342,379,
%T A348338 418,459,502,547,594,643,694,747,802,859,918,979,1042,1107,1174,1243,
%U A348338 1314,1387,1462,1539,1618,1699,1782,1867,1954,2043,2134,2227,2322,2419,2518
%N A348338 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 -> 2*m.
%C A348338 For n >= 1, the largest number of steps required is 4*5^(n-1) + n.
%F A348338 For n >= 1, a(n) = a(n-1) + 2*n - ceiling(log_5 ((n+1)/16)), or a(n) = n^2 + n + 2 - Sum_{2..n} ceiling(log_5 ((i+1)/16)).
%e A348338 a(1) = 4. As shown below, integers ending with 0, 5, {2, 4, 6 or 8}, and {1, 3, 7, or 9} require 1, 2, 4, and 5 steps to repeat the last digit, respectively. Therefore, the distinct numbers of steps are {1, 2, 4, 5} and a(1) = 4.
%e A348338 _ 1=>2===>4<=7
%e A348338 v \ ^ v
%e A348338 5==>0== 3=>6<===8<=9
%e A348338 a(2) = 9 because the distinct steps are {1, 2, 3, 4, 5, 6, 20, 21, 22}, as shown by the paths of the last two digits of integers.
%e A348338 _ 1,51 27,77 29,79 33,83 41,91 7,57
%e A348338 v \ v v v v v v
%e A348338 25,75==>50==>0== 2 54 58 66 82 14
%e A348338 v v v v v v
%e A348338 4=====>8=====>16====>32====>64====>28
%e A348338 5,55 35,85 ^ v
%e A348338 v v 13,63=>26=>52 56<=78<=39,89
%e A348338 10 70 ^ v
%e A348338 v v 19,69=>38=>76 12<==6<==3,53
%e A348338 20==>40 ^ v
%e A348338 ^ v 47,97=>94=>88 24<=62<=31,81
%e A348338 60<==80 ^ v
%e A348338 ^ ^ 11,61=>22=>44 48<=74<=37,87
%e A348338 30 90 ^ v
%e A348338 ^ ^ 72<====36<====68<====84<====92<====96
%e A348338 15,65 45,95 ^ ^ ^ ^ ^ ^
%e A348338 86 18 34 42 46 98
%e A348338 ^ ^ ^ ^ ^ ^
%e A348338 43,93 9,59 17,67 21,71 23,73 49,99
%e A348338 a(3) = 15 because the distinct steps for n = 3 are {1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 100, 101, 102, 103}.
%o A348338 (Python)
%o A348338 def tail(m):
%o A348338 global n; s = str(m)
%o A348338 return m if len(s) <= n else int(s[-n:])
%o A348338 for n in range(1, 9):
%o A348338 M = []
%o A348338 for i in range(10**n):
%o A348338 t = i; L = [t]
%o A348338 while i >= 0:
%o A348338 t = tail(2*t)
%o A348338 if t not in L: L.append(t)
%o A348338 else: break
%o A348338 d = len(L)
%o A348338 if d not in M: M.append(d)
%o A348338 print(len(M), end = ', ')
%o A348338 (Python)
%o A348338 def A348338(n):
%o A348338 m, s = 10**n, set()
%o A348338 for k in range(m):
%o A348338 c, k2, kset = 0, k, set()
%o A348338 while k2 not in kset:
%o A348338 kset.add(k2)
%o A348338 c += 1
%o A348338 k2 = 2*k2 % m
%o A348338 s.add(c)
%o A348338 return len(s) # _Chai Wah Wu_, Oct 19 2021
%o A348338 (PARI) a(n) = n + (n+1)*(n-1-t=(logint(5*(n+1)>>4+(n<3), 5))) + 4*5^t - (2-n)*(n<3); \\ _Jinyuan Wang_, Nov 02 2021
%Y A348338 Cf. A005843, A348339.
%K A348338 nonn,base
%O A348338 0,2
%A A348338 _Ya-Ping Lu_, Oct 13 2021
%E A348338 a(9)-a(10) from _Martin Ehrenstein_, Oct 20 2021
%E A348338 a(0) prepended and a(11)-a(14) from _Martin Ehrenstein_, Oct 29 2021
%E A348338 More terms from _Jinyuan Wang_, Nov 02 2021