A087975 a(n) is the smallest number having n as the largest number of distinct prime factors of any term in its 3x+1 trajectory.

1, 2, 3, 15, 210, 703, 20895, 239263, 4497567, 28203751, 2222250495, 103612115103

Offset: 0

Author

Labos Elemer, Sep 25 2003

Keywords

• nonn
• hard
• more

Comments

From Kevin P. Thompson, Jan 10 2022: (Start)

If C(k) is the k-th term of the Collatz trajectory of a(n), the following are the first terms of the Collatz trajectories that achieve the required factorization.

n=0: C(1) = 1 =

n=1: C(1) = 2 = 2

n=2: C(2) = 10 = 2*5

n=3: C(4) = 70 = 2*5*7

n=4: C(1) = 210 = 2*3*5*7

n=5: C(87) = 93940 = 2^2*5*7*11*61

n=6: C(48) = 3257170 = 2*5*7*19*31*79

n=7: C(32) = 8731030 = 2*5*7*11*17*23*29

n=8: C(97) = 163908677842 = 2*7^2*13*19*23*37*73*109

n=9: C(71) = 31688082790 = 2*5*7*13*17*23*29*37*83

n=10: C(81) = 113760010350580 = 2^2*5*11*17*19*37*43*47*79*271

n=11: C(64) = 68985183404830 = 2*5*7*11*17*19*23*31*43*83*109

(End)

Examples

			a(5) = 703 because 703 is the smallest number whose Collatz trajectory has 5 as the largest number of distinct prime factors of any term in its 3x+1 trajectory; its trajectory includes 93940 = 2^2 * 5 * 7 * 11 * 61.
		

Links

• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A087974.

Programs

Mathematica

Join[{1},Table[k=0;While[Last@NestWhileList[If[OddQ@#,3#+1,#/2]&,++k,#!=1&&PrimeNu@#!=n&]==1];k,{n,5}]] (* Giorgos Kalogeropoulos, Jan 11 2022 *)

Extensions

a(6)-a(11) from Kevin P. Thompson, Jan 10 2022