A253389 - OEIS

%I #27 May 11 2024 21:32:56

%S 13625124998637487500,28428684442602686000,1652983470,2,31975,4400,

%T 61964512293803548770,8264462800,0,1234567890,14233809528576619047,

%U 16969012743858543270,9412305876,27,15937,18125674943632987050,2376,1652983470,0,24680,84530839221546916077,22644848642280060680,27,2,7,22840808842260464660,28556013027144398697,2488420660,0

%N a(n) is the repeating digit pattern in penultimate digit of successive powers of n (omitting initial powers without at least two digits).

%C This is looking one step past the well-known rules for the last digit of successive powers:  Powers of integers ending in digit 2 always repeat 2486 in the last digit pattern, powers of integers ending in digit 3 always repeat 3971, of integers ending in 4 repeat 46, of integers ending in 1, 5, and 6 repeat themselves, of integers ending in 7 repeat 7931, of integers ending in 8 repeat 8426, and of integers ending in 9 repeat 91.

%C Is there a pattern in the repeating patterns in the penultimate digits?  Possibly 99 patterns, for x = 01 to 99?

%C All pattern lengths are a divisor of 20, as n^2 == n^22 (mod 100). - _Walter Roscello_, Jan 22 2023

%C Some terms have quasi-periodic patterns with first nonzero digit(s) not in the period, such as a(14), a(15) and a(18) ignoring first digit 1 while a(22) ignoring first digit 2. In these cases, the periodic patterns of a(n) could be rewritten by rotation. - _Lerong Zhu_, May 10 2024

%e Powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096...

%e Second-to-the-last digits, skipping the one-digit powers: 1,3,6,2,5,1,2,4,9,9,8,6,3,7,4,8,7,5,0,0,...

%e Find repeating pattern and concatenate digits: 13625124998637487500

%e 10 does not repeat its penultimate digit (1), so a(10)=0.

%Y Cf. A160590 (penultimate digit of 2^n).

%K nonn,base,easy

%O 2,1

%A _Erik Maher_, Dec 31 2014

%E Missing a(8) inserted by _Walter Roscello_, Jan 22 2023

%E a(12)-a(30) from _Lerong Zhu_, May 10 2024