A181666 Numbers whose odd part is of the form (4^k-1)/3.

1, 2, 4, 5, 8, 10, 16, 20, 21, 32, 40, 42, 64, 80, 84, 85, 128, 160, 168, 170, 256, 320, 336, 340, 341, 512, 640, 672, 680, 682, 1024, 1280, 1344, 1360, 1364, 1365, 2048, 2560, 2688, 2720, 2728, 2730, 4096, 5120, 5376, 5440, 5456, 5460, 5461, 8192, 10240

Offset: 1

Author

Ralf Stephan, Nov 18 2010

Keywords

• nonn

Comments

Also numbers that need at most one (3x+1) operation to reach 1 in the Collatz conjecture.

Also, terms of A023758 divisible by 3, divided by 3 (conjectured).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
• Index entries for sequences related to 3x+1 (or Collatz) problem.

Crossrefs

Cf. A000265, A023758, A055086, A082375, A000975, A321873.

Programs

Mathematica

Select[Range[2600], IntegerQ@ Log[4, 3 # + 1] &@ NestWhile[#/2 &, #, IntegerQ[#/2] &] &] (* Michael De Vlieger, May 12 2017 *)
seq[max_] := Module[{kmax = Floor[Log[4, 3*max+1]], s = {}, s1, odd},Do[odd = (4^k-1)/3; s1 = 2^Range[0, Floor[Log2[max/odd]]] * odd; s = Join[s, s1], {k, 1, kmax}]; Union[s]]; seq[10240] (* Amiram Eldar, Aug 31 2024 *)

PARI

for(n=2, 2000, o=3*n/2^valuation(n,2)+1; b=ispower(o); if(b&&round(sqrtn(o,b\2))==4, print1(n, ", ")))

Python

for a in range(2, 17):
    print()
    for b in range(a - 2, -1, -2):
        print(((1 << a) - (1 << b)) // 3, end=", ")
# André Hallqvist, Jul 25 2019
# This script shows the sequence as an irregular table whose main diagonal is A000975, called the "Lichtenberg sequence" by Andreas M. Hinz. - Peter Luschny, Jul 10 2022

Formula

a(n) = (4 * 2^A055086(n-1) - 2^A082375(n-1))/3. - André Hallqvist, Jul 17 2019

Sum_{n>=1} 1/a(n) = 2 * A321873 = 2.5265861162... . - Amiram Eldar, Jul 19 2022