A333861 The sum of the Hamming weights of the elements of the Collatz orbit of n.

1, 2, 11, 3, 7, 13, 35, 4, 43, 9, 29, 15, 16, 38, 43, 5, 24, 45, 49, 11, 10, 32, 35, 17, 58, 19, 527, 41, 42, 47, 507, 6, 66, 26, 28, 47, 50, 52, 100, 13, 520, 13, 73, 35, 34, 39, 497, 19, 59, 61, 66, 22, 21, 531, 537, 44, 85, 46, 91, 51, 52, 512, 523, 7, 67

Offset

1,2

Links

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

Formula

a(n) = Sum_{k=1..A008908(n)} A000120(A070165(n,k)). - Alois P. Heinz, Apr 10 2020

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 sum is a(3) = 11.

Mathematica

a[n_] := Total[DigitCount[#, 2, 1] & /@ NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, # > 1 &]]; Array[a, 65] (* 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 += hammingweight(n));
c;
}

Crossrefs

Cf. A000120, A008908, A070165, A333860.

Sequence in context: A342043 A351997 A104662 * A338845 A121713 A357820

Adjacent sequences: A333858 A333859 A333860 * A333862 A333863 A333864

Keyword

nonn,base,easy,look

Author

Markus Sigg, Apr 08 2020

Status

approved