A226687 -id:A226687 - cp's OEIS frontend

A226625 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x-k cycle associated with A226623(n).

1, 3, 11, 4, 6, 6, 17, 19, 19, 19, 19, 19, 19, 19, 19, 34, 12, 9, 5, 22, 22, 22, 12, 17, 17, 17, 69, 7, 7, 7, 18, 44, 22, 38, 38, 38, 38, 38, 22, 22, 33, 33, 22, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 48, 12

Offset: 1

Geoffrey H. Morley, Jun 13 2013

Conjecture: Every cycle with the same value of k (k>1) has the same proportion of odd and even elements. Thus if n>1 then A226626(n)/A226625(n) has the same value for each m where A226628(n) <= m < A226628(n+1).

			The irregular array starts:
(k=1)  1, 3, 11;
(k=11) 4;
(k=17) 6, 6;
(k=19) 17;
a(4)=4 is the length of the 3x-11 cycle {19,23,29,38} associated with A226623(4)=19.

Geoffrey H. Morley, Rows 1..280 of array, flattened

Row n begins with a(A226628(n)) and has length A226629(n). k=A226630(n)
The cycle associated with a(n) has A226626(n) odd elements of which A226624(n) is the largest.
Cf. A226609, A226627, A226631, A226686, A226687, A226688.

A226686 Conjectured record-breaking lengths, for ascending positive integers k, of primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

11, 17, 19, 34, 69, 84, 85, 168, 171, 176, 179, 228, 252, 285

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.
For primitive cycles, GCD(k,6)=1.

k = A226687(n). The smallest integer in the T_k cycle(s) associated with a(n) is A226688(n).

Cf. A226625, A226670.

A226688 Smallest positive integer in the primitive Collatz-like 3x-k cycle(s) with the conjectured record-breaking length A226686(n).

Original entry on oeis.org

17, 115, 2263, 113, 769, 436099, 6073, 3735185, 194197, 230891, 4575823, 823027, 21111517, 12477095

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

k = A226687(n).

Cf. A226672.

A226671 Integer k associated with the conjectured record-breaking length A226670(n) of primitive Collatz-like 3x+k cycles.

Original entry on oeis.org

1, 5, 17, 23, 29, 61, 85, 107, 125, 139, 143, 197, 253, 313, 371, 509, 563, 1135, 1163, 1307, 1699, 3299, 8431, 11491, 16819, 22097, 24917, 49787, 67475, 76733, 99391, 110273, 111611, 144379, 273641, 308219

Offset: 1

Geoffrey H. Morley, Jun 16 2013

nonn

Cf. A226672, A226687.

A226689 Conjectured record-breaking numbers of odd elements, for ascending positive integers k, in primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

Original entry on oeis.org

7, 11, 12, 22, 44, 53, 54, 106, 108, 112, 113, 144, 159, 180

Offset: 1

Geoffrey H. Morley, Jun 19 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.
For primitive cycles, GCD(k,6)=1.
For n<15, but probably not for all n, k = A226687(n) and the smallest integer in the T_k cycle associated with a(n) is A226688(n).

Cf. A226626, A226673.

A229122 For odd m, let f(m) be the odd part of 3m+1. a(n) is the least positive number of f-iterations of 2n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2

Offset: 1

Vladimir Shevelev, Oct 07 2013

nonn

Since 1 is odious number, the conjecture that all a(n) > 0 is a very weak form of the "3x+1" (Collatz) conjecture.

We conjecture that this sequence is unbounded.

			For n = 26, 2*n - 1 = 51; f(51) = 77 is evil; f(77) = 29 is evil; f(29) = 11 is odious, so a(26) = 3.

Cf, A226686, A226687.

Table[m = 2 n - 1; NestWhile[# + 1 &, 1, !OddQ[DigitCount[m = # / 2^IntegerExponent[#, 2] & [3 m + 1], 2][[1]]] &], {n, 100}] (* Peter J. C. Moses, Oct 13 2013 *)

More terms from Peter J. C. Moses

cp's OEIS Frontend

A226625 Irregular array read by rows. a(n) is the length of the primitive Collatz-like 3x-k cycle associated with A226623(n).

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A226686 Conjectured record-breaking lengths, for ascending positive integers k, of primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

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A226688 Smallest positive integer in the primitive Collatz-like 3x-k cycle(s) with the conjectured record-breaking length A226686(n).

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A226671 Integer k associated with the conjectured record-breaking length A226670(n) of primitive Collatz-like 3x+k cycles.

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A226689 Conjectured record-breaking numbers of odd elements, for ascending positive integers k, in primitive cycles of positive integers under iteration by the Collatz-like 3x-k function.

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A229122 For odd m, let f(m) be the odd part of 3*m+1. a(n) is the least positive number of f-iterations of 2*n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.

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A229122 For odd m, let f(m) be the odd part of 3m+1. a(n) is the least positive number of f-iterations of 2n-1 to reach an odious number (A000069), or 0 if no such number of f-iterations exists.