A198584 Odd numbers producing 3 odd numbers in the Collatz (3x+1) iteration.

3, 13, 53, 113, 213, 227, 453, 853, 909, 1813, 3413, 3637, 7253, 7281, 13653, 14549, 14563, 29013, 29125, 54613, 58197, 58253, 116053, 116501, 218453, 232789, 233013, 464213, 466005, 466033, 873813, 931157, 932053, 932067, 1856853, 1864021, 1864133

Offset

1,1

Comments

One of the odd numbers is always 1. So besides a(n), there is one other odd number, A198585(n), which is a term in A002450.

Sequences A228871 and A228872 show that there are two sequences here: the odd numbers in order and out of order. - T. D. Noe, Sep 12 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..1009

Formula

Numbers of the form (2^m*(2^n-1)/3-1)/3 where n == 2 (mod 6) if m is even and n == 4 (mod 5) if m is odd. - Charles R Greathouse IV, Sep 09 2022

Example

The Collatz iteration of 113 is 113, 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2, 1, which shows that 113, 85, and 1 are the three odd terms.

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {}; Do[If[Length[Select[Collatz[n], OddQ]] == 3, AppendTo[t, n]], {n, 1, 10000, 2}]; t

Prog

(Python)
# get n-th term in sequence
def isqrt(n):
  i=0
  while(i*i<=n):
    i+=1
  return i-1
for n in range (200):
  s = isqrt(3*n)//3
  a = s*3
  b = (a*a)//3
  c = n-b
  d = 4*(n*3+a+(c<s)+(c>4*s+1)+(c>5*s+1))+5
  e = isqrt(d)
  f = e-1-( (d-e*e) >> 1 )
  r = ((((8<<e)-(1<<f))//3)-1)//3
  print(r, end =", ") # André Hallqvist, Jul 25 2019
(Python)
# just prints the sequence
for a in range (5, 100, 1):
  for b in range(a-8+4*(a&1), 0, -6):
    print(( ((1<<a)-(1<<b))//3-1)//3 , end=", ") # André Hallqvist, Aug 14 2019

Crossrefs

Cf. A062053 (numbers producing 3 odds in their Collatz iteration).

Cf. A092893 (least number producing n odd numbers).

Cf. A198586-A198593 (odd numbers producing 2-10 odd numbers).

Cf. A228871, A228872.

Sequence in context: A346409 A082376 A065059 * A346382 A342815 A072197

Adjacent sequences: A198581 A198582 A198583 * A198585 A198586 A198587

Keyword

nonn,easy

Author

T. D. Noe, Oct 28 2011

Status

approved