A155072 Positive integers n such that the base-2 MR-expansion of 1/n is periodic with period (n-1)/4.

17, 41, 97, 137, 193, 313, 401, 409, 449, 521, 569, 761, 769, 809, 857, 929, 977, 1009, 1129, 1297, 1361, 1409, 1489, 1697, 1873, 1993, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713, 2729, 2753, 2777, 2801, 2897, 3001

Offset

1,1

Comments

See A136042 for the definition of the MR-expansion of a positive real number.

It appears that all terms of this sequence are primes of the form 8n+1 (A007519).

Apparently a subsequence of A115591. - Mia Boudreau, Jun 17 2025

Links

Mia Boudreau, Table of n, a(n) for n = 1..1000

Example

Applying the definition of the base-2 MR-expansion to 1/17 gives 1/17 -> 2/17 -> 4/17 -> 8/17 -> 16/17 -> 32/17 -> 15/17 -> 30/17 -> 13/17 -> 26/17 -> 9/17 -> 18/17 -> 1/17 -> ..., which shows that the expansion begins {5,1,1,1,...} and has period 4=(17-1)/4.

Mathematica

a[p_] := 1 + Sum[2 Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];
Select[Range[500], Reduce[a[#]^2 == 2 # + 1, Integers] &];
2 % + 1 (* Gerry Martens, May 01 2016 *)

Crossrefs

Cf. A136042, A007519, A115591, A136043.

Sequence in context: A281792 A070179 A293394 * A243451 A145991 A139961

Adjacent sequences: A155069 A155070 A155071 * A155073 A155074 A155075

Keyword

nonn

Author

John W. Layman, Jan 19 2009

Extensions

More terms from Mia Boudreau, Jun 17 2025

Status

approved