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A371074
Number of the rightmost decimal digits of n that are the same as those of n^n.
3
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, 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, 0, 1, 1, 2, 0, 1, 0, 2, 2, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1
OFFSET
0,12
COMMENTS
The common digits might include leading 0's (such as at n = 51 or n = 57) and they are included in the total.
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.
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).
LINKS
Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.
FORMULA
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)).
EXAMPLE
a(0) = 0 since 0^0 = 1 so that 0 and 0^0 have no digits in common.
For n = 51, a(n) = 3 since 51^51 == 5051 (mod 10^4).
KEYWORD
nonn,base
AUTHOR
Marco Ripà, Mar 10 2024
STATUS
approved