Geoffrey H. Morley - cp's OEIS frontend

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

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)}

A226615 Length of the Collatz-like 3x+k cycle associated with A226614(n).

Original entry on oeis.org

2, 3, 6, 4, 7, 5, 20, 11, 12, 28, 6, 38, 44, 48, 72, 18, 106, 29, 75, 7, 37, 14, 10, 42, 72, 66, 10, 68, 38, 58, 72, 8, 43, 110, 22, 33, 68, 29, 42, 71, 13, 46, 121, 28, 182, 200, 47, 9, 21, 60, 108, 28, 156, 19, 22, 85, 79, 151, 62, 56, 71, 60, 78, 226, 104, 192

Offset: 1

Geoffrey H. Morley, Aug 02 2013

nonn

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

PARI
```
See A226614.
```

a(n) = A226609(A226612(floor(A226614(n)/3)+1)).

For n>227, this formula requires terms beyond A226609(10000).

A226677 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

1, 1, 11, 17, 115, 31, 1, 29, 1417, 371, 19, 23, 8977, 77, 431, 2465, 2069, 3299, 193, 451, 139, 25, 5233, 131, 1739, 10993, 3037, 121, 7061, 11329, 9479, 145, 2425, 46199, 1871, 217, 3551, 26183, 14083, 26281, 7237, 605, 181, 113, 3299, 11431, 119773, 2465

Offset: 1

Geoffrey H. Morley, Jul 05 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.
Conjecture: a(n)>0 for all n.

Geoffrey H. Morley, Table of n, a(n) for n = 1..60

Cf. A226626, A226661, A226676.

A226676 Smallest positive integer k (or 0 if no such k) with a primitive cycle of n positive integers under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

1, 0, 1, 11, 49, 17, 115, 473, 31, 791, 1, 29, 11491, 371, 641, 2167, 19, 119, 23, 3211, 106537, 77, 431, 2465, 2069, 5575

Offset: 1

Geoffrey H. Morley, Jul 05 2013

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.
Conjecture: For n>2, a(n)>0.

Cf. A226625, A226660, A226677.

A226661 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 7, 5, 25, 13, 59, 47, 11, 29, 145, 59, 31, 115, 79, 13, 47, 5, 17, 125, 79, 263, 49, 169, 91, 191, 23, 601, 323, 193, 109, 311, 73, 149, 265, 571, 95, 491, 697, 695, 137, 29, 119, 383, 575, 283, 121, 263, 233, 163, 193, 283, 479, 107, 203, 437, 85, 491, 349

Offset: 1

Geoffrey H. Morley, Jul 05 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.
Conjecture: a(n)>0 for all n.

Geoffrey H. Morley, Table of n, a(n) for n = 1..2520

Cf. A226610, A226660, A226677.

A226660 Smallest positive integer k with 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, 7, 5, 11, 17, 13, 97, 59, 19, 55, 233, 11, 73, 25, 29, 47, 215, 41, 103, 145, 31, 13, 119, 131, 5, 47, 53, 67, 17, 337, 125, 115, 485, 133, 127, 49, 119, 191, 293, 133, 23, 79, 103, 191, 167, 91, 409, 329, 217, 109, 449, 241, 361, 353, 1303, 239, 149, 73

Offset: 2

Geoffrey H. Morley, Jul 05 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.

Geoffrey H. Morley, Table of n, a(n) for n = 2..3908

Cf. A226609, A226661, A226676.

A226621 Irregular array read by rows in which a(n) is the number of primitive '3x+k' cycles associated with A226619(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 1, 3, 1, 5, 5, 2, 1, 1, 3, 7, 1, 2, 5, 7, 3, 1, 1, 4, 8, 1, 1, 14, 14, 8, 4, 1, 1, 3, 12, 1, 6, 2, 1, 1, 14, 23, 20, 12, 2, 1, 1, 5, 15, 29, 1, 1, 1, 2, 5, 5, 42, 34, 25, 14, 5, 1, 1, 5, 16, 28, 2, 9, 1, 1, 1

Offset: 1

Geoffrey H. Morley, Jul 02 2013

The terms in row n are associated with cycles of length n (see A226619).

			The irregular array starts:
1, 1;
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 1, 3, 1, 1; ...

Geoffrey H. Morley, Rows 1..26 of array, flattened

Cf. A226620.

A226620 Irregular array read by rows in which a(n) is the number of odd elements in each primitive '3x+k' cycle associated with A226619(n).

Original entry on oeis.org

1, 0, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 2, 3, 2, 1, 6, 5, 2, 4, 3, 2, 1, 7, 6, 5, 4, 4, 4, 3, 2, 1, 8, 7, 6, 6, 3, 5, 4, 3, 2, 1, 9, 8, 7, 8, 6, 5, 2, 2, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 7, 5, 3, 5, 4, 5, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 9, 8, 8, 8, 8, 4, 4, 7

Offset: 1

Geoffrey H. Morley, Jul 02 2013

The terms in row n are associated with cycles of length n (see A226619).

			The irregular array starts:
1, 0;
1;
2, 1;
3, 2, 1;
4, 3, 2, 1;
5, 4, 2, 3, 2, 1; ...

Geoffrey H. Morley, Rows 1..26 of array, flattened

Cf. A226621.

A226617 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, n of which are odd including 1, under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 11, 43, 55, 643, 97, 673, 41, 1843, 329, 59, 113, 5603, 289, 6505, 77, 407, 127, 499, 79, 865, 749

Offset: 1

Geoffrey H. Morley, Jul 03 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.
Conjecture: a(n)>0 for all n.

			The cycle associated with a(1)=1 is {1,2}, with a(2)=11 is {1,7,16,8,4,2}, and with a(3)=43 is {1,23,56,28,14,7,32,16,8,4,2}.

Cf. A226607, A226610, A226616.

A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

-1, 1, 1, -1, 5, -11, 7, 13, -49, 5, 23, 29, -179, -17, 11, 37, 55, 61, -601, -115, 17, 47, 101, 119, 125, -1931, -473, 13, 25, 35, 175, 229, 247, 253, -6049, -1675, -217, -31, 97, 269, 431, 485, 503, 509, -18659, -5537, -1163, -791, 59, 71, 145, 203, 295, 781, 943, 997, 1015, 1021

Offset: 1

Geoffrey H. Morley, Jul 02 2013

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.
We associate the cycle {0} with k = A226606(2) = 1.
For n>1 the first term of row n is 2^n-3^(n-1), and the last term is A036563(n) = 2^n-3.

			The irregular array starts:
-1, 1;
1;
-1, 5;
-11, 7, 13;
-49, 5, 23, 29; ...

Geoffrey H. Morley, Rows 1..26 of array, flattened

Cf. A226605, A226606, A226618, A226620, A226621.

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User: Geoffrey H. Morley

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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A226615 Length of the Collatz-like 3x+k cycle associated with A226614(n).

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A226677 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x-k function.

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A226676 Smallest positive integer k (or 0 if no such k) with a primitive cycle of n positive integers under iteration by the Collatz-like 3x-k function.

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A226661 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, exactly n of which are odd, under iteration by the Collatz-like 3x+k function.

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A226660 Smallest positive integer k with a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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A226621 Irregular array read by rows in which a(n) is the number of primitive '3x+k' cycles associated with A226619(n).

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A226620 Irregular array read by rows in which a(n) is the number of odd elements in each primitive '3x+k' cycle associated with A226619(n).

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A226617 Smallest positive integer k (or 0 if no such k) with a primitive cycle of positive integers, n of which are odd including 1, under iteration by the Collatz-like 3x+k function.

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A226619 Irregular array read by rows in which row n lists the integers k, in ascending order, for which there is a primitive cycle of n positive integers under iteration by the Collatz-like 3x+k function.

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