A033493 Sum of the numbers in the trajectory of n for the 3x+1 problem.

1, 3, 49, 7, 36, 55, 288, 15, 339, 46, 259, 67, 119, 302, 694, 31, 214, 357, 519, 66, 148, 281, 633, 91, 658, 145, 101440, 330, 442, 724, 101104, 63, 841, 248, 540, 393, 535, 557, 2344, 106, 101331, 190, 1338, 325, 497, 679, 100979, 139, 806, 708, 1130, 197

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• look

Comments

Given a power of two, the value in this sequence is the next higher Mersenne number, or a(2^m) = 2^(m + 1) - 1. - Alonso del Arte, Apr 10 2009

Conjecture: a(n) = A006577(n)^2 only at a(3) = 49. Verified for n <= 10^7. - Luca Santarsiero, Jul 13 2025

Examples

			a(5) = 36 because the Ulam's conjecture trajectory sequence starting on 5 runs 5, 16, 8, 4, 2, 1 and therefore 5 + 16 + 8 + 4 + 2 + 1 = 36. - _Alonso del Arte_, Apr 10 2009
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Collatz Problem
• Wikipedia, Collatz conjecture
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Apart from initial term, exactly the same as A049074.

Cf. A006370, A006577, A070165.

Programs

Haskell

a033493 = sum . a070165_row  -- Reinhard Zumkeller, Oct 08 2011

Maple

a:= proc(n) option remember; n+`if`(n=1, 0,
      a(`if`(n::even, n/2, 3*n+1)))
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 29 2021

Mathematica

collatz[1] = 1; collatz[n_Integer?OddQ] := 3n + 1; collatz[n_Integer?EvenQ] := n/2; Table[-1 + Plus @@ FixedPointList[collatz, n], {n, 60}] (* Alonso del Arte, Apr 10 2009 *)

Python

def a(n):
    if n==1: return 1
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        l+=[n, ]
        if n<2: break
    return sum(l)
print([a(n) for n in range(1, 101)])  # Indranil Ghosh, Apr 14 2017

Formula

a(n) = Sum_{k=1..A006577(n)} A070165(k). - Reinhard Zumkeller, Oct 08 2011

Extensions

Corrected a(16) to 31 to match other powers of 2; removed duplicate value of a(48) = 139 because a(49) = 806 and not 139. - Alonso del Arte, Apr 10 2009