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

Author

Markus Sigg, Apr 08 2020

Keywords

• nonn
• base
• easy
• look

Examples

			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.
		

Links

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

Crossrefs

Cf. A000120, A006370, A008908, A070165, A333860.

Programs

Mathematica

a[n_] := Total[DigitCount[#, 2, 1] & /@ NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, # > 1 &]]; Array[a, 65] (* Amiram Eldar, Jul 29 2023 *)

PARI

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

Formula

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

a(n) = a(A006370(n)) + A000120(n), with a(1) = 1. - Alan Michael Gómez Calderón, Jul 15 2025