A381707 Smallest initial value for unimodal Collatz (3x+1)/2 glide sequence that begins with exactly n increases.

5, 3, 23, 15, 95, 575, 383, 255, 5631, 25599, 104447, 69631, 745471, 3293183, 2195455, 12648447, 97910783, 65273855, 43515903, 1460666367, 6700400639, 4466933759, 71697432575, 47798288383, 764873277439, 1242923270143, 3760646520831, 8371159695359, 5580773130239, 3720515420159

Offset: 1

Author

David Dewan, Mar 04 2025

Keywords

• nonn

Comments

A unimodal Collatz glide sequence is successive rises x -> (3x+1)/2 followed by successive falls x -> x/2 until dropping below its starting x.

After n increases, there are ceiling(n*log(3)/log(2) - n) decreases to drop below the initial value.

Examples

			For n=3, the smallest starting x = a(3) = 23 has trajectory
  23 - 35 -> 53 ->  80  -> 40 -> 20
     \-----------/      \------/
     n=3 increases   decreases to < initial
		

Links

• David Dewan, Unimodal Glide Sequences

Crossrefs

Cf. A126241, A122437, A122458.

Programs

Mathematica

a[n_]:=2^n ModularInverse[3^n,2^Max[Ceiling[Log2[3^n]-n],2]]-1; Array[a,30]

PARI

a(n)={my(m=2^(logint(3^n,2) - n + 1 + (n==1))); 2^n*lift(1/Mod(3^n,m)) - 1} \\ Andrew Howroyd, Mar 09 2025

Formula

a(n) = 2^n * (3^(-n) mod 2^max(2, ceiling(log2(3^n)-n))) - 1.