A371074 Number of the rightmost decimal digits of n that are the same as those of n^n.

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

Author

Marco Ripà, Mar 10 2024

Keywords

• nonn
• base

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).

Examples

			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).
		

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.

Crossrefs

Cf. A000312, A082576, A317905, A349425, A369826, A369624, A370211.

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)).