A259207 -id:A259207 - cp's OEIS frontend

A057688 Trajectory of 5 under the '5x+1' map.

5, 26, 13, 66, 33, 11, 56, 28, 14, 7, 36, 18, 9, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6, 3, 1, 6

Offset: 0

N. J. A. Sloane, Oct 20 2000

The 'Px + 1 map': if x is divisible by any prime less than P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1. This is similar to A057684, but with P = 5 instead of P = 13. - Alonso del Arte, Jul 04 2015

			7 is odd and not divisible by 3, so it's followed by 5 * 7 + 1 = 36.
36 is even, so it's followed by 36/2 = 18.
18 is even, so it's followed by 18/2 = 9.
9 is odd and divisible by 3, so it's followed by 9/3 = 3.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057684, A057685, A057686, A057687, A057689, A057690, A057691.

Cf. A259207.

NestList[If[EvenQ[#], #/2, If[Mod[#, 3] == 0, #/3, 5# + 1]] &, 5, 100] (* Alonso del Arte, Jul 04 2015 *)

Vec((5 + 26*x + 13*x^2 + 61*x^3 + 7*x^4 - 2*x^5 - 10*x^6 - 5*x^7 + 3*x^8 - 49*x^9 + 8*x^10 + 4*x^11 + 2*x^12 - 33*x^13 - 17*x^14 - 3*x^15) / ((1 - x)*(1 + x + x^2)) + O(x^100)) \\ Colin Barker, Oct 10 2019

a(0) = 5, a(n) = a(n - 1)/2 if a(n - 1) is even, a(n) = a(n - 1)/3 if a(n - 1) is odd and divisible by 3, a(n) = 5a(n - 1) otherwise.
From Colin Barker, Oct 10 2019: (Start)
G.f.: (5 + 26*x + 13*x^2 + 61*x^3 + 7*x^4 - 2*x^5 - 10*x^6 - 5*x^7 + 3*x^8 - 49*x^9 + 8*x^10 + 4*x^11 + 2*x^12 - 33*x^13 - 17*x^14 - 3*x^15) / ((1 - x)*(1 + x + x^2)).
a(n) = a(n-3) for n>15.
(End)

A328010 The 5x + 1 sequence beginning at 17.

Original entry on oeis.org

17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17

Offset: 0

Antoine Beaulieu, Oct 01 2019

The 5x+1 problem is similar to the 3x+1 or Collatz problem. For some starting values it is known that the 5x+1 trajectory will tend to infinity or enter a periodic orbit.
Alex V. Kontorovich & Jeffrey C. Lagarias conjectured that there are very few periodic orbits. One of them is shown here.
The two other known periodic orbits are given in the crossrefs.

Colin Barker, Table of n, a(n) for n = 0..1000
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems arXiv:0910.1944 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A259207, A328011.

Vec((17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Oct 05 2019

a(n+1) = 5*a(n) + 1 if a(n) is odd, a(n+1) = a(n)/2 otherwise.
From Colin Barker, Oct 04 2019: (Start)
G.f.: (17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-10) for n>9.
(End)

A328011 The 5x + 1 sequence beginning at 1.

Original entry on oeis.org

1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1

Offset: 0

Antoine Beaulieu, Oct 01 2019

See A328010 for further information.

Colin Barker, Table of n, a(n) for n = 0..1000
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A259207, A328010.

Vec((1 + 6*x + 3*x^2 + 16*x^3 + 8*x^4 + 4*x^5 + 2*x^6) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^80)) \\ Colin Barker, Oct 08 2019

a(n+1) = 5*a(n) + 1 if a(n) is odd, a(n+1) = a(n)/2 otherwise.
From Colin Barker, Oct 08 2019: (Start)
G.f.: (1 + 6*x + 3*x^2 + 16*x^3 + 8*x^4 + 4*x^5 + 2*x^6) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-7) for n>6.
(End)

A028389 The 5x + 1 sequence beginning at 7.

Original entry on oeis.org

7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, 7166, 3583, 17916, 8958, 4479, 22396, 11198, 5599, 27996, 13998, 6999, 34996, 17498, 8749, 43746, 21873, 109366, 54683, 273416, 136708, 68354, 34177, 170886, 85443, 427216

Offset: 0

N. J. A. Sloane, Dean Hickerson

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A259207, A328010, A328011.

f := proc(n) option remember; if n = 0 then 7 elif f(n-1) mod 2 = 0 then f(n-1)/2 else 5*f(n-1)+1; fi; end; seq(f(n), n=0..50);

a[0] = 7; a[n_] := a[n] = If[OddQ[a[n-1]], 5 a[n-1] + 1, a[n-1]/2];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 03 2020 *)
NestList[If[OddQ[#],5#+1,#/2]&,7,50] (* Harvey P. Dale, Nov 03 2022 *)

a(0) = 7; a(n) = 5*a(n-1) + 1 if a(n-1) is odd, a(n) = a(n-1)/2 otherwise.

Named edited by Andrew Howroyd, Aug 21 2020

A259193 5x + 1 sequence beginning at 11.

Original entry on oeis.org

11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, 7166, 3583, 17916, 8958, 4479, 22396, 11198, 5599, 27996, 13998, 6999, 34996, 17498, 8749, 43746, 21873, 109366, 54683, 273416, 136708, 68354, 34177, 170886, 85443

Offset: 0

Alonso del Arte, Jun 21 2015

			11 is odd, so it's followed by 5 * 11 + 1 = 56.
56 is even, so it's followed by 56/2 = 28.

Alonso del Arte, Table of n, a(n) for n = 0..999
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems arXiv:0910.1944 [math.NT], 2009.

Cf. A259207, A057688.

[n eq 1 select 11 else IsOdd(Self(n-1)) select 5*Self(n-1)+1 else Self(n-1) div 2: n in [1..80]]; // Vincenzo Librandi, Jul 04 2015

NestList[If[EvenQ[#], #/2, 5# + 1] &, 5, 100]

a(0) = 11; a(n) = 5a(n - 1) + 1 if a(n - 1) is odd, a(n) = a(n - 1)/2 otherwise.

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

A271623 a(0)=7; a(n) = 7*a(n-1) + 1 if a(n-1) is odd, a(n) = a(n-1)/2 otherwise.

Original entry on oeis.org

7, 50, 25, 176, 88, 44, 22, 11, 78, 39, 274, 137, 960, 480, 240, 120, 60, 30, 15, 106, 53, 372, 186, 93, 652, 326, 163, 1142, 571, 3998, 1999, 13994, 6997, 48980, 24490, 12245, 85716, 42858, 21429, 150004, 75002, 37501, 262508, 131254, 65627, 459390, 229695

Offset: 0

Vincenzo Librandi, Apr 11 2016

Conjectured: No term of the sequence is equal to a previous term.

			7 is odd, so it is followed by 7*7 + 1 = 50.
50 is even, so it is followed by 50/2 = 25.

Cf. A033478, A259207.

[n eq 1 select 7 else IsOdd(Self(n-1)) select 7*Self(n-1)+1 else Self(n-1) div 2: n in [1..60]];

NestList[If[EvenQ[#], #/2, 7 # + 1]&, 7, 60]

Collatz(n, lim=0) = {my(c=n, e=0, L=List(n)); if(lim==0, e=1; lim=n*10^2);
for(i=1, lim, if(c%2==0, c=c/2, c=7*c+1); listput(L, c); if(e&&c==1, break)); return(Vec(L));}
print(Collatz(7)) \\ Adapted from A008884 \\ Altug Alkan, Apr 11 2016

A305057 -5x + 1 sequence starting at 5.

Original entry on oeis.org

5, -24, -12, -6, -3, 16, 8, 4, 2, 1, -4, -2, -1, 6, 3, -14, -7, 36, 18, 9, -44, -22, -11, 56, 28, 14, 7, -34, -17, 86, 43, -214, -107, 536, 268, 134, 67, -334, -167, 836, 418, 209, -1044, -522, -261, 1306, 653, -3264, -1632, -816, -408, -204, -102, -51, 256, 128, 64, 32, 16, 8, 4, 2, 1, -4

Offset: 0

Alonso del Arte, May 26 2018

			5 is odd, so it's followed by (-5) * 5 + 1 = -25 + 1 = -24.
-14 is even, so it's followed by -14/2 = -7.

Cf. A259207.

[n eq 1 select 5 else IsOdd(Self(n-1)) select -5*Self(n-1)+1 else Self(n-1) div 2: n in [1..80]]; // Vincenzo Librandi, Jun 11 2018

NestList[If[EvenQ[#], #/2, -5# + 1] &, 5, 100]

a(0) = 5, a(n) = (-5)*a(n - 1) + 1 if a(n - 1) is odd, a(n) = a(n - 1)/2 otherwise.

cp's OEIS Frontend

A057688 Trajectory of 5 under the '5x+1' map.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328010 The 5x + 1 sequence beginning at 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A328011 The 5x + 1 sequence beginning at 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A028389 The 5x + 1 sequence beginning at 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A259193 5x + 1 sequence beginning at 11.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A271623 a(0)=7; a(n) = 7*a(n-1) + 1 if a(n-1) is odd, a(n) = a(n-1)/2 otherwise.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A305057 -5x + 1 sequence starting at 5.

Views

Author