A349767 - OEIS

A349767

Numbers m such that 2^m - m is divisible by 5.

3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57, 63, 74, 76, 77, 83, 94, 96, 97, 103, 114, 116, 117, 123, 134, 136, 137, 143, 154, 156, 157, 163, 174, 176, 177, 183, 194, 196, 197, 203, 214, 216, 217, 223, 234, 236, 237, 243, 254, 256, 257, 263, 274, 276, 277, 283, 294, 296, 297, 303

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 5.

Equivalently, numbers that are congruent to {3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57} mod 60, <==> numbers that are congruent to {+-3, +-14, +-16, +-17, +-23, +-34} mod 60.

REFERENCES

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

LINKS

Table of n, a(n) for n=1..61.

The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.

Index to sequences related to Olympiads.

MAPLE

filter:= n -> 2^n-n mod 5 = 0 : select(filter, [$1..400]);

MATHEMATICA

Select[Range[300], PowerMod[2, #, 5] == Mod[#, 5] &] (* Amiram Eldar, Dec 10 2021 *)

PROG

(PARI) isok(m) = Mod(2, 5)^m == Mod(m, 5); \\ Michel Marcus, Dec 10 2021

(Python)

def ok(n): return pow(2, n, 5) == n%5