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Positive integers n such that the base-2 MR-expansion of 1/n is periodic with period (n-1)/4.
3

%I #7 May 06 2016 12:02:08

%S 17,41,97,137,193,313,401,409,449,521,569,761,769,809,857,929,977

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

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

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

%e 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.

%t a[p_] := 1 + Sum[2 Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];

%t Select[Range[500], Reduce[a[#]^2 == 2 # + 1, Integers] &];

%t 2 % + 1 (* _Gerry Martens_, May 01 2016 *)

%Y Cf. A007519, A136042, A136043.

%K nonn,more

%O 1,1

%A _John W. Layman_, Jan 19 2009