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

1,3

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)

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A003095, A064434.

Sequence in context: A277741 A241138 A241349 * A111934 A098509 A019910

Adjacent sequences: A330402 A330403 A330404 * A330406 A330407 A330408

Keyword

easy,nonn

Author

Matthew Ryan, Dec 12 2019

Status

approved