A226607 -id:A226607 - cp's OEIS frontend

A226609 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x+k cycle associated with A226607(n).

2, 3, 5, 5, 27, 27, 4, 6, 14, 4, 24, 8, 8, 8, 8, 8, 8, 8, 7, 31, 11, 5, 5, 43, 16, 8, 5, 17, 65, 65, 23, 8, 8, 6, 6, 6, 20, 11, 18, 28, 7, 7, 7, 7, 7, 38, 29, 12, 6, 28, 28, 10, 10, 10, 10, 10, 10, 6, 66, 24, 30, 10, 10, 27, 27, 27, 27, 27, 12, 60, 15, 38

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  2;
(k=5)  3, 5, 5, 27, 27;
(k=7)  4;
(k=11) 6, 14;
a(2)=3 is the length of the 3x+5 cycle {1,4,2} associated with A226607(2)=1.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has A226610(n) odd elements of which A226608(n) is the largest.
Cf. A226611, A226625, A226670-A226672.

A226612 Index of the first element of row n of A226607.

Original entry on oeis.org

1, 2, 7, 8, 10, 19, 21, 22, 25, 27, 31, 32, 34, 37, 38, 39, 46, 47, 48, 51, 58, 60, 61, 62, 69, 72, 73, 77, 80, 81, 82, 85, 88, 90, 97, 99, 100, 101, 102, 104, 109, 111, 115, 117, 120, 122, 127, 128, 131, 134, 136, 138, 139, 140, 144, 146, 149, 151, 153, 160

Offset: 1

Geoffrey H. Morley, Jun 13 2013

nonn

Geoffrey H. Morley, Table of n, a(n) for n = 1..6667

a(n) is also the index of the first element of row n in A226608 to A226611.

Cf. A226613, A226628.

A226610 Irregular array read by rows. a(n) is the number of odd elements in the primitive 3x+k cycle associated with A226607(n).

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 2, 2, 8, 1, 15, 5, 5, 5, 5, 5, 5, 5, 2, 18, 5, 2, 2, 26, 8, 4, 1, 9, 41, 41, 12, 4, 4, 3, 3, 3, 8, 3, 7, 16, 4, 4, 4, 4, 4, 22, 17, 4, 2, 16, 11, 6, 6, 6, 6, 6, 6, 1, 41, 12, 16, 5, 5, 17, 17, 17, 17, 17, 4, 32, 8, 16, 20, 20, 14, 14

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 3, 3, 17, 17;
(k=7)  2;
(k=11) 2, 8;
a(2)=1 is the number of odd elements in the 3x+5 cycle {1,4,2} associated with A226607(2)=1.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has length A226609(n) and its largest element is A226608(n).
Cf. A226611, A226626.

A226608 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x+k cycle associated with A226607(n).

Original entry on oeis.org

1, 1, 49, 37, 2773, 3397, 11, 7, 79, 1, 1853, 1121, 797, 665, 905, 653, 761, 557, 5, 181, 35, 19, 11, 1651, 137, 41, 1, 121, 2277097, 1051393, 131, 127, 79, 89, 53, 65, 157, 23, 43, 643, 331, 223, 211, 259, 175, 1409, 757, 71, 19, 827, 139, 1399, 775, 751, 967, 559, 571

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 49, 37, 2772, 3397;
(k=7)  11;
(k=11) 7, 79;
a(3)=49 is the largest element in the 3x+5 cycle {19,31,49,76,38} associated with A226607(3)=19.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has length A226609(n) and A226610(n) odd elements.
Cf. A226611, A226624, A226667-A226669.

A226611 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226607(n), where T_k is the 3x+k function associated with A226607(n).

Original entry on oeis.org

1, 1, 3, 23, 123, 171, 1, 1, 3, 1, 19, 99, 147, 163, 123, 283, 159, 319, 1, 9, 1, 5, 7, 1, 1, 3, 1, 3, 2531, 5859, 1, 1, 3, 1, 3, 7, 1, 1, 5, 1, 9, 33, 39, 21, 101, 1, 1, 1, 7, 9, 1, 3, 149, 21, 93, 125, 221, 1, 175, 1, 1, 1, 7, 2585, 1073, 2301, 4121, 893

Offset: 1

Geoffrey H. Morley, Jun 13 2013

			The irregular array starts:
(k=1)  1;
(k=5)  1, 3, 23, 123, 171;
(k=7)  1;
(k=11) 1, 3;
a(3)=3 is the smallest starting value for a 3x+5 trajectory that includes A226607(3)=19. The trajectory is {3,7,13,22,11,19,...}.

Geoffrey H. Morley, Rows 1..2032 of array, flattened

Row n begins with a(A226612(n)) and has length A226613(n).
The cycle associated with a(n) has length A226609(n) and A226610(n) odd elements of which A226608(n) is the largest.
Cf. A226627.

A226613 a(n) is the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x+k function, where n=floor(k/3)+1.

Original entry on oeis.org

1, 5, 1, 2, 9, 2, 1, 3, 2, 4, 1, 2, 3, 1, 1, 7, 1, 1, 3, 7, 2, 1, 1, 7, 3, 1, 4, 3, 1, 1, 3, 3, 2, 7, 2, 1, 1, 1, 2, 5, 2, 4, 2, 3, 2, 5, 1, 3, 3, 2, 2, 1, 1, 4, 2, 3, 2, 2, 7, 1, 3, 1, 2, 3, 4, 1, 2, 2, 1, 4, 1, 3, 2, 1, 2, 1, 8, 19, 3, 4, 2, 2, 6, 2, 3, 3, 7, 3

Offset: 1

Geoffrey H. Morley, Jun 13 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.

Geoffrey H. Morley, Table of n, a(n) for n = 1..6667
E. G. Belaga and M. Mignotte, Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem, Preprint math. 2000/17, Univ. Louis Pasteur, Strasbourg (2000). [Table 1 on page 19 gives a(1) to a(500).]

a(n) is the number of terms in the n-th row of A226607 to A226611.

Cf. A226629, A226663.

a(n) = A226612(n+1) - A226612(n).

A226623 Irregular array read by rows in which row n lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the Collatz-like 3x-k function, where k = A226630(n).

Original entry on oeis.org

1, 5, 17, 19, 65, 73, 115, 2263, 2359, 2743, 2963, 3091, 3415, 3743, 4819, 113, 109, 95, 65, 989, 1153, 1165, 293, 511, 505, 625, 769, 211, 227, 251, 311, 1085, 2089, 7471, 10883, 13963, 15875, 16099, 1291, 1355, 1367, 1495, 1931, 2059, 2123, 2203, 2219, 2251

Offset: 1

Geoffrey H. Morley, Jun 13 2013

The 3x-k function T_k is defined by T_k(x) = x/2 if x is even, (3x-k)/2 if x is odd, where k is odd.

Lagarias (1990) called a T_k cycle primitive if its elements are all relatively prime to k or, equivalently, if its elements are not a common multiple of the elements of another cycle.

			The irregular array starts:
(k=1)  1, 5, 17;
(k=11) 19;
(k=17) 65, 73;
(k=19) 115;
a(4)=19 is the smallest number in the 3x-11 cycle {19,23,29,38}.

Geoffrey H. Morley, Rows 1..280 of array, flattened
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53.

Row n begins with a(A226628(n)) and has length A226629(n).
The smallest starting value whose trajectory includes a(n) is A226627(n). The cycle associated with a(n) has length A226625(n) and A226626(n) odd elements of which A226624(n) is the largest
Cf. A226607, A226681, A226682.

A226614 Positive integers k for which 1 is in a cycle of integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 11, 13, 17, 29, 41, 43, 55, 59, 61, 77, 79, 91, 95, 97, 107, 113, 119, 125, 127, 137, 145, 155, 185, 193, 203, 209, 215, 239, 247, 253, 257, 275, 281, 289, 317, 329, 335, 353, 355, 407, 437, 445, 473, 493, 499, 509, 553, 559, 593, 629, 637, 643, 673, 697

Offset: 1

Geoffrey H. Morley, Aug 02 2013

nonn

The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd. GCD(k,6)=1.
When k=2^m-3, T_k has a cycle containing 1. Hence the sequence is infinite.
a(n) is in the sequence if and only if A226607(A226612(floor(a(n)/3)+1)) = 1.
Trivially, members of the sequence are not divisible by 2 or 3. Of the first 10^4 members, only 1,066 are squareful, which is about one third of the expected density. - Ralf Stephan, Aug 05 2013

Geoffrey H. Morley, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A226615-A226618.

\\ 5.5 hours (2.33 Ghz Intel Core 2)
{k=1; n=1;
until(n>10000, x=1; y=1; len=0;
  until(x==y, if(x%2==0, x=x/2, x=(3*x+k)/2);
    if(y%2==0, y=y/2, y=(3*y+k)/2);
    if(y%2==0, y=y/2, y=(3*y+k)/2); len++);
  if(x==1, write("b226614.txt",n," ",k);
    write("b226615.txt",n," ",len); n++);
  k+=(k+3)%6)}

A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 13, 29, 11, 17, 253, 509, 145, 43, 55, 355, 137, 1129, 1007, 131069, 97, 643, 41, 553, 281, 8388605, 4069, 4793489, 3817, 1843, 59, 113, 1301, 2155, 9397, 289, 131153, 3247, 949, 127, 77

Offset: 2

Geoffrey H. Morley, Jul 02 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.
For n>1, T_k has a primitive cycle of length n which includes 1 when k = A036563(n) = 2^n-3. So a(n) <= 2^n-3.

Cf. A226607, A226609, A226617, A226618.

A226665 Conjectured record-breaking maximal values, for ascending positive integers k, of the minimal elements of the primitive cycles of positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 347, 7055, 177337, 212665, 219913, 379541, 413803, 822535, 1391321, 8013899, 21619279, 21834347, 28306063, 37550317, 168536521, 189763177

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.

			a(1)=1 because {1,2}, with minimal element 1, is the only known '3x+1' cycle of positive integers.
k=5 is the next value of k>1 with GCD(k,6)=1. The minimal element in each of the five known primitive '3x+5' cycles of positive integers is 1, 19, 23, 187 and 347. 347>a(1) so a(2)=347.

k = A226666(n).

Cf. A226607, A226681.

cp's OEIS Frontend

A226609 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x+k cycle associated with A226607(n).

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A226612 Index of the first element of row n of A226607.

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A226610 Irregular array read by rows. a(n) is the number of odd elements in the primitive 3x+k cycle associated with A226607(n).

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A226608 Irregular array read by rows. a(n) is the largest element in the primitive Collatz-like 3x+k cycle associated with A226607(n).

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A226611 Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226607(n), where T_k is the 3x+k function associated with A226607(n).

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A226613 a(n) is the conjectured number of primitive cycles of positive integers under iteration by the Collatz-like 3x+k function, where n=floor(k/3)+1.

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Formula

A226623 Irregular array read by rows in which row n lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the Collatz-like 3x-k function, where k = A226630(n).

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A226614 Positive integers k for which 1 is in a cycle of integers under iteration by the Collatz-like 3x+k function.

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PARI

A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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A226665 Conjectured record-breaking maximal values, for ascending positive integers k, of the minimal elements of the primitive cycles of positive integers under iteration by the Collatz-like 3x+k function.

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