A226618 -id:A226618 - cp's OEIS frontend

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.

-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.

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.

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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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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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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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