A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 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

Offset: 1

Author

T. D. Noe, May 28 2010

Keywords

• nonn
• easy

Comments

The odd non-multiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz pre-image. The next odd number in the Collatz iteration of a(2n) is 6n-1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each non-multiple of 3, there are an infinite number of odd numbers in its Collatz pre-image. For example:

Odd pre-images of 1: 1, 5, 21, 85, 341,... (A002450)

Odd pre-images of 5: 3, 13, 53, 213, 853,... (A072197)

Odd pre-images of 7: 9, 37, 149, 597, 2389,...

Odd pre-images of 11: 7, 29, 117, 469, 1877,...(A072261)

In each case, the pre-image sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of pre-images is in A178415.

a(n) = A047529(P(n)), with the permutation P(n) = A006368(n-1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely. - Wolfdieter Lang, Sep 21 2021

Links

• Matthew House, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A002450, A006368, A007310, A022998, A047529, A072197, A072261, A114752.

Programs

Mathematica

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

Formula

a(n) = (n - 1)*(3 - (-1)^n) + 1. [Bogart B. Strauss, Sep 20 2013, adapted to the offset by Matthew House, Feb 14 2017]

From Matthew House, Feb 14 2017: (Start)

G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1 - x)^2*(1 + x)^2).

a(n) = 2*a(n-2) - a(n-4). (End)

From Philippe Deléham, Nov 06 2023: (Start)

a(2*n) = 4*n-1, a(2*n+1) = 8*n+1.

a(n) = 2*A022998(n-1)+1.

a(n) = 2*A114752(n)-1. (End)