A369771 Number of the rightmost decimal digits of n^(n^n) that are the same as those of n^(n^(n^n)).

1, 0, 1, 1, 2, 2, 8, 4, 4, 2, 3, 10000000000, 4, 2, 3, 2, 13, 4, 3, 2, 3, 104857600000000000000000000, 4, 1, 2, 4, 12, 8, 2, 2, 3, 205891132094649000000000000000000000000000000, 4, 4, 3, 2, 7, 4, 3, 1, 3, 12089258196146291747061760000000000000000000000000000000000000000

Offset

-1,5

Comments

The common digits might include leading 0's (such as at n = 5) 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*(n^n) which is all the rightmost 0's of n^(n^n).

Links

Table of n, a(n) for n=-1..40.

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) == n^(n^(n^n)) (mod 10^(a(n))) and n^(n^n) <> n^(n^(n^n)) (mod 10^(a(n)+1)).

Example

a(-1) = 1 since (-1)^(-1) = -1 (which is a one-digit number);
a(0) = 0 since 0^0 = 1 so that 0^(0^0) = 0 and 0^(0^(0^0)) = 1 have no digits in common.
For n=5, a(n)=8 since 5^(5^5) == 908203125 (mod 10^9) while 5^(5^(5^5)) == 408203125 (mod 10^9).

Crossrefs

Cf. A002488, A002489, A317905, A349425, A369624.

Sequence in context: A195361 A274041 A049331 * A239677 A331333 A120399

Adjacent sequences: A369768 A369769 A369770 * A369772 A369773 A369774

Keyword

sign,base

Author

Marco Ripà, Jan 31 2024

Status

approved