A079665 -id:A079665 - cp's OEIS frontend

A079581 Consider pairs (r,s) such that the polynomial (x^r+1) divides (x^s+1) and 1 <= r < s. This sequence gives the s values; A079673 gives the r values.

3, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 15, 15, 17, 18, 18, 19, 20, 21, 21, 21, 22, 23, 24, 25, 25, 26, 27, 27, 27, 28, 29, 30, 30, 30, 31, 33, 33, 33, 34, 35, 35, 35, 36, 36, 37, 38, 39, 39, 39, 40, 41, 42, 42, 42, 43, 44, 45, 45, 45, 45, 45, 46, 47, 48, 49, 49, 50, 50, 51

Offset: 1

Jose R. Brox (tautocrona(AT)terra.es), Jan 25 2003

nonn

(x^r+1) divides (x^s+1) iff s/r is an odd integer.

			9 is in the sequence twice because (x^1+1) and (x^3+1) divide (x^9+1).

Cf. A079665, A079672, A079673.

Edited by Don Reble, Jun 12 2003

A079672 Numbers of the form (3^s+1)/(3^r+1) for s > 1, 1 <= r <= s-1.

Original entry on oeis.org

7, 61, 73, 547, 4921, 703, 5905, 44287, 6481, 398581, 478297, 3587227, 512461, 58807, 32285041, 38742049, 530713, 290565367, 42521761, 2615088301, 373584043, 4780783, 3138105961, 23535794707, 43040161, 211822152361, 3472494301

Offset: 1

Jose R. Brox (tautocrona(AT)terra.es), Jan 25 2003

nonn

(b^s+1) / (b^r+1) is an integer iff s/r is odd. - Jose Brox (tautocrona(AT)terra.es), Dec 27 2005

Cf. A079665, A079581, A079673.

for(x=2,26, for(y=1,x-1,if(Mod(2^x+1,2^y+1),0,print1((3^x+1)/(3^y+1)",")))) \\ The Mod(2^x+1,2^y+1) is not a bug, since the exponents do not depend on the base in which they are calculated.

A079673 Consider pairs (r,s) such that the polynomial (x^r+1) divides (x^s+1) and 1 <= r < s. This sequence gives the r values; A079581 gives the s values.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 1, 3, 5, 1, 2, 6, 1, 4, 1, 3, 7, 2, 1, 8, 1, 5, 2, 1, 3, 9, 4, 1, 2, 6, 10, 1, 1, 3, 11, 2, 1, 5, 7, 4, 12, 1, 2, 1, 3, 13, 8, 1, 2, 6, 14, 1, 4, 1, 3, 5, 9, 15, 2, 1, 16, 1, 7, 2, 10, 1, 3, 17, 4, 1, 2, 6, 18, 1, 5, 11, 8, 1, 3, 19, 2, 1, 4, 12, 20, 1, 2, 1, 3, 7, 9, 21, 1

Offset: 1

Jose R. Brox (tautocrona(AT)terra.es), Jan 25 2003

nonn

(x^r+1) divides (x^s+1) iff s/r is an odd integer.

			a(5)=1 and a(6)=3 because A079581(5)=A079581(6)=9 and (x^1+1) and (x^3+1) divide (x^9+1).

Cf. A079581, A079665, A079672.

Edited by Don Reble, Jun 12 2003

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

Original entry on oeis.org

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

Thomas Ordowski, Feb 16 2024

nonn

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.

			(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.

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

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

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

More terms from Michel Marcus, Feb 17 2024

cp's OEIS Frontend

A079581 Consider pairs (r,s) such that the polynomial (x^r+1) divides (x^s+1) and 1 <= r < s. This sequence gives the s values; A079673 gives the r values.

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A079672 Numbers of the form (3^s+1)/(3^r+1) for s > 1, 1 <= r <= s-1.

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A079673 Consider pairs (r,s) such that the polynomial (x^r+1) divides (x^s+1) and 1 <= r < s. This sequence gives the r values; A079581 gives the s values.

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

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PARI

Extensions