

A178414


Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.


3



1, 3, 9, 7, 17, 11, 25, 15, 33, 19, 41, 23, 49, 27, 57, 31, 65, 35, 73, 39, 81, 43, 89, 47, 97, 51, 105, 55, 113, 59, 121, 63, 129, 67, 137, 71, 145, 75, 153, 79, 161, 83, 169, 87, 177, 91, 185, 95, 193, 99, 201, 103, 209, 107, 217, 111, 225, 115, 233, 119, 241, 123, 249
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OFFSET

1,2


COMMENTS

The odd nonmultiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz preimage. The next odd number in the Collatz iteration of a(2n) is 6n1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each nonmultiple of 3, there are an infinite number of odd numbers in its Collatz preimage. For example:
Odd preimages of 1: 1, 5, 21, 85, 341,... (A002450)
Odd preimages of 5: 3, 13, 53, 213, 853,... (A072197)
Odd preimages of 7: 9, 37, 149, 597, 2389,...
Odd preimages of 11: 7, 29, 117, 469, 1877,...(A072261)
In each case, the preimage sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of preimages is in A178415.
a(n) = A047529(P(n)), with the permutation P(n) = A006368(n1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely.  Wolfdieter Lang, Sep 21 2021


LINKS



FORMULA

G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1  x)^2*(1 + x)^2).
a(n) = 2*a(n2)  a(n4). (End)
a(2*n) = 4*n1, a(2*n+1) = 8*n+1.


MATHEMATICA

Riffle[1+8*Range[0, 50], 3+4*Range[0, 50]]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



