A189510 Digital root of n^n.

1, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 02 2011

Keywords

• nonn
• base
• easy

Comments

a(n) = A010888(A000312(n)).

For n >= 1, this sequence is periodic with period 18. The sequence repeats [1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8,9]. - Nathaniel Johnston, May 04 2011

Links

• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Crossrefs

Cf. A010888, A030132, A145389.

Programs

Maple

A189510 := proc(n) return ((n^n-1) mod 9) + 1: end: seq(A189510(n), n=0..80); # Nathaniel Johnston, May 04 2011

Mathematica

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[n^n]], {n,0,200}]
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9},86]] (* Ray Chandler, Aug 27 2015 *)
PadRight[{1},100,{9,1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8}] (* Harvey P. Dale, Jul 31 2025 *)

Python

def A189510(n): return (9,1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8)[n%18] if n else 1 # Chai Wah Wu, Feb 09 2023

Formula

From Chai Wah Wu, Feb 09 2023: (Start)

a(n) = a(n-18) for n > 18.

G.f.: (-8*x^18 - 8*x^17 - 7*x^16 - 9*x^15 - 7*x^14 - 4*x^13 - 9*x^12 - 5*x^11 - x^10 - 9*x^9 - x^8 - 7*x^7 - 9*x^6 - 2*x^5 - 4*x^4 - 9*x^3 - 4*x^2 - x - 1)/(x^18 - 1). (End)

Extensions

a(0) corrected by Reinhard Zumkeller, May 03 2011