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A370425 Integers of the form (2^x + 1) / (2^y + 1). 3

%I #8 Feb 18 2024 11:38:04

%S 1,3,11,13,43,57,171,205,241,683,993,2731,3277,3641,4033,10923,16257,

%T 43691,52429,61681,65281,174763,233017,261633,699051,838861,1016801,

%U 1047553,2796203,4192257,11184811,13421773,14913081,15790321,16519105,16773121,44739243,67100673,178956971

%N Integers of the form (2^x + 1) / (2^y + 1).

%C The integers k for which the equation 2^x - k = k*2^y - 1 has a solution x,y > 0.

%C If x,y > 0, then 2^y + 1 divides 2^x + 1 if and only if x/y is odd.

%C The prime numbers of this sequence are A281728.

%H Max Alekseyev, <a href="/A370425/b370425.txt">Table of n, a(n) for n = 1..1000</a>

%e (2^5+1)/(2^1+1) = 11 = 1011,

%e (2^10+1)/(2^2+1) = 205 = 11001101,

%e (2^15+1)/(2^3+1) = 3641 = 111000111001,

%e (2^20+1)/(2^4+1) = 61681 = 1111000011110001,

%e (2^25+1)/(2^5+1) = 1016801 = 11111000001111100001,

%e (2^30+1)/(2^6+1) = 16519105 = 111111000000111111000001,

%e (2^35+1)/(2^7+1) = 266354561 = 1111111000000011111110000001, ...

%e Note that all the above examples are A020518(n) for n > 0.

%o (PARI) get_xy(m) = my(x, y, t); y=valuation(m-1, 2); t=m*(2^y+1)-1; if(t!=2^(x=valuation(t, 2)), [], [x, y]); \\ _Max Alekseyev_, Feb 18 2024

%Y Cf. A064896 (integers of the form (2^x-1)/(2^y-1)), A079665, A281728.

%K nonn

%O 1,2

%A _Thomas Ordowski_, Feb 16 2024

%E More terms from _Michel Marcus_, Feb 17 2024

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Last modified June 22 15:06 EDT 2024. Contains 373587 sequences. (Running on oeis4.)