A358348 Numbers k such that k == k^k (mod 9).

1, 4, 7, 9, 10, 13, 16, 17, 18, 19, 22, 25, 27, 28, 31, 34, 35, 36, 37, 40, 43, 45, 46, 49, 52, 53, 54, 55, 58, 61, 63, 64, 67, 70, 71, 72, 73, 76, 79, 81, 82, 85, 88, 89, 90, 91, 94, 97, 99, 100, 103, 106, 107, 108, 109, 112, 115, 117, 118, 121, 124, 125, 126

Offset

1,2

Comments

Each multiple of 9 is in the sequence. Additionally, the squares are also present.

References

M. Fujiwara and Y. Ogawa, Introduction to Truly Beautiful Mathematics. Tokyo: Chikuma Shobo, 2005.

Links

Table of n, a(n) for n=1..63.

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

Formula

G.f.: x*(x+1)*(x^7+3*x^5+x^3+x^2+2*x+1)/((1-x)^2*(1+x^3+x^6)*(1+x+x^2)). - Alois P. Heinz, Feb 08 2023

a(n) = 2*(n+1) - b(n) where b(n>=0) = 2,3,2,1,1,2,1,0,1,2,3,2,... has period 9. - Kevin Ryde, Mar 26 2023

Example

4 is a term since 4^4 = 256 == 4 (mod 9).

Maple

A358348 := proc(n)
    2*(n+1)-op(modp(n, 9)+1, [2, 3, 2, 1, 1, 2, 1, 0, 1]) ;
end proc:
seq(A358348(n), n=1..50) ; # R. J. Mathar, Mar 29 2023

Mathematica

Select[Range[130], MemberQ[{0, 1, 4, 7, 9, 10, 13, 16, 17}, Mod[#, 18]] &] (* Amiram Eldar, Nov 12 2022 *)

Prog

(PARI) isok(k) = k == Mod(k, 9)^k; \\ Michel Marcus, Nov 22 2022
(Python)
def A358348(n):
    return ((0, 1, 4, 7, 9, 10, 13, 16, 17)[m := n % 9]
         + (n - m << 1))  # Chai Wah Wu, Feb 09 2023

Crossrefs

Cf. A007953, A010888, A082576, A189510.

Sequence in context: A045752 A266410 A010380 * A056991 A242660 A010389

Adjacent sequences: A358345 A358346 A358347 * A358349 A358350 A358351

Keyword

nonn,base,easy

Author

Ivan Stoykov, Nov 11 2022

Status

approved