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

1, 3, 11, 13, 43, 57, 171, 205, 241, 683, 993, 2731, 3277, 3641, 4033, 10923, 16257, 43691, 52429, 61681, 65281, 174763, 233017, 261633, 699051, 838861, 1016801, 1047553, 2796203, 4192257, 11184811, 13421773, 14913081, 15790321, 16519105, 16773121, 44739243, 67100673, 178956971

Offset

1,2

Comments

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

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

The prime numbers of this sequence are A281728.

Links

Max Alekseyev, Table of n, a(n) for n = 1..1000

Example

(2^5+1)/(2^1+1) = 11 = 1011,
(2^10+1)/(2^2+1) = 205 = 11001101,
(2^15+1)/(2^3+1) = 3641 = 111000111001,
(2^20+1)/(2^4+1) = 61681 = 1111000011110001,
(2^25+1)/(2^5+1) = 1016801 = 11111000001111100001,
(2^30+1)/(2^6+1) = 16519105 = 111111000000111111000001,
(2^35+1)/(2^7+1) = 266354561 = 1111111000000011111110000001, ...
Note that all the above examples are A020518(n) for n > 0.

Prog

(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

Crossrefs

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

Sequence in context: A181086 A274343 A113049 * A225097 A079665 A281728

Adjacent sequences: A370422 A370423 A370424 * A370426 A370427 A370428

Keyword

nonn

Author

Thomas Ordowski, Feb 16 2024

Extensions

More terms from Michel Marcus, Feb 17 2024

Status

approved