A330405 a(1) = 0; thereafter a(n) = (a(n-1)^2+1) mod n.

0, 1, 2, 1, 2, 5, 5, 2, 5, 6, 4, 5, 0, 1, 2, 5, 9, 10, 6, 17, 17, 4, 17, 2, 5, 0, 1, 2, 5, 26, 26, 5, 26, 31, 17, 2, 5, 26, 14, 37, 17, 38, 26, 17, 20, 33, 9, 34, 30, 1, 2, 5, 26, 29, 17, 10, 44, 23, 58, 5, 26, 57, 37, 26, 27, 4, 17, 18, 49, 22, 59, 26, 20, 31, 62, 45, 24

Offset: 1

Author

Matthew Ryan, Dec 12 2019

Keywords

• easy
• nonn

Comments

Does the value 0 appear infinitely many times? - Rémy Sigrist, Dec 16 2019

From Michael De Vlieger, Jan 26 2020: (Start)

Observations based on a(n) for 1 <= n <= 300000:

The value 0 appears at indices n = {1, 13, 26, 89, 205, 530, 2041, 276205, ...}.

The value 1 appears at indices n = {2, 4, 14, 27, 50, 90, 99, 175, 188, 206, 531, 2042, 5445, 6845, 7200, 18225, 24389, 25215, 37538, 46875, 48672, 53066, 79527, 93900, 147875, 176267, 186576, 196025, 254457, 276206, ...}. Let M be the indices in a(n) where 1 appears.

The subsequence {1, 2, 5, 26} appears with the first term at index n = 27, and apparently for all subsequent indices listed n M.

The subsequence {1, 2, 5, 26, 677} appears with the first term at index n = 2042, and apparently for all subsequent indices listed n M.

A stable next term in the subsequence S = {1, 2, 5, 26, 677} is not yet apparent, given 300000 terms of a(n). (End)

Examples

			a(1) = 0; a(2) = (0^2+1) mod 2 = 1; a(3) = (1^2+1) mod 2 = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003095, A064434.

Programs

Mathematica

Nest[Append[#1, Mod[#1[[#2 - 1]]^2 + 1, #2]] & @@ {#, Length@ # + 1} &, {0}, 76] (* Michael De Vlieger, Dec 16 2019 *)

PARI

v=0; for (n=1, 77, print1 (v=(v^2+1)%n", ")) \\ Rémy Sigrist, Dec 16 2019

Formula

a(1) = 0; a(n) = (a(n-1)^2+1) mod n.