A350053 -id:A350053 - cp's OEIS frontend

A198584 Odd numbers producing exactly 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

T. D. Noe, Oct 28 2011

One of the odd numbers is always 1. So besides a(n), there is exactly 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
Start with the numbers in A350053. If k is in sequence then so is 4*k+1. - Ralf Stephan, Jun 18 2025

			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.

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

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.

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

# 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+(c4*s+1)+(c>5*s+1))+5
  e = isqrt(d)
  f = e-1-( (d-e*e) >> 1 )
  r = ((((8<André Hallqvist, Jul 25 2019

# just prints the sequence
for a in range (5,100,1):
  for b in range(a-8+4*(a&1),0,-6):
    print(( ((1<André Hallqvist, Aug 14 2019

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

a(n) = (16*2^floor(b(n)) - 2^(2*floor((b(n) - 1)/2) + 3*floor(b(n)) - 6*(floor(b(n)/2) - floor((floor(b(n))^2 + 20)/12) + n) - 2))/9 - 1/3 where b(n) = sqrt(3)*sqrt(4*n - 3). - Alan Michael Gómez Calderón, Feb 02 2025

A350054 a(n) = (4^(3*n+2) - 7)/9.

Original entry on oeis.org

113, 7281, 466033, 29826161, 1908874353, 122167958641, 7818749353073, 500399958596721, 32025597350190193, 2049638230412172401, 131176846746379033713, 8395318191768258157681, 537300364273168522091633, 34387223313482785413864561

Offset: 1

Wolfdieter Lang, Jan 20 2022

Bisection of A350053, namely the even part. The odd part is given in A228871.

Winston de Greef, Table of n, a(n) for n = 1..551
Index entries for linear recurrences with constant coefficients, signature (65,-64).

Cf. A133853, A228871, A350053.

Table[(4^(3*n + 2) - 7)/9, {n, 1, 14}] (* Amiram Eldar, Jan 21 2022 *)

a(n) = (4^(3*n+2) - 7)/9 \\ Winston de Greef, Jan 27 2024

a(n) = (2^(6*n+4) - 7)/9, n >= 1.
G.f.: x*(113 - 64*x)/((1 - x)*(1 - 64*x)).
a(n) = 112*A133853(n) + 1. - Hugo Pfoertner, Jan 21 2022

cp's OEIS Frontend

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

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