A333860 The maximum Hamming (binary) weight of the elements of the Collatz orbit of n, or -1 if 1 is never reached.

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 3, 3, 4, 1, 3, 3, 4, 2, 3, 3, 4, 2, 4, 3, 8, 3, 4, 4, 8, 1, 4, 3, 4, 3, 3, 4, 5, 2, 8, 3, 4, 3, 4, 4, 8, 2, 3, 4, 4, 3, 4, 8, 8, 3, 4, 4, 5, 4, 5, 8, 8, 1, 3, 4, 4, 3, 3, 4, 8, 3, 8, 3, 4, 4, 4, 5, 6, 2, 5, 8, 8, 3, 4, 4, 5

Offset

1,3

Links

Markus Sigg, Table of n, a(n) for n = 1..10000

Formula

a(n) = max(A000120(n), A352895(n)) = max(A000120(n), a(A006370(n))). - Antti Karttunen, Apr 10 2022

Example

The Collatz orbit of 3 is 3,10,5,16,8,4,2,1. The Hamming weights are 2,2,2,1,1,1,1,1. The maximum is a(3) = 2.

Mathematica

a[n_] := Max[DigitCount[#, 2, 1] & /@ NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, # > 1 &]]; Array[a, 100] (* Amiram Eldar, Jul 29 2023 *)

Prog

(PARI) a(n) = {
my(c = hammingweight(n));
while(n>1, n = if(n%2 == 0, n/2, 3*n+1); c = max(c, hammingweight(n)));
c;
}

Crossrefs

Cf. A000120, A006370, A008908, A070165, A333861, A352895, A352897.

Sequence in context: A355735 A355733 A256757 * A277314 A120562 A178692

Adjacent sequences: A333857 A333858 A333859 * A333861 A333862 A333863

Keyword

nonn,easy,base

Author

Markus Sigg, Apr 08 2020

Extensions

Escape clause added to the definition by Antti Karttunen, Apr 10 2022

Status

approved