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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
1, 2, 3, 15, 210, 703, 20895, 239263, 4497567, 28203751, 2222250495, 103612115103
OFFSET
0,2
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 = <no prime factorization exists>
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)
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A087974.
Sequence in context: A297421 A180723 A162102 * A162103 A162104 A162105
KEYWORD
nonn,hard,more
AUTHOR
Labos Elemer, Sep 25 2003
EXTENSIONS
a(6)-a(11) from Kevin P. Thompson, Jan 10 2022
STATUS
approved