A033493 -id:A033493 - cp's OEIS frontend

A270814 a(1)=0; thereafter a(2k)=k+a(k), a(2k+1)=6k+4+a(6k+4).

0, 1, 46, 3, 31, 49, 281, 7, 330, 36, 248, 55, 106, 288, 679, 15, 197, 339, 500, 46, 127, 259, 610, 67, 633, 119, 101413, 302, 413, 694, 101073, 31, 808, 214, 505, 357, 498, 519, 2305, 66, 101290, 148, 1295, 281, 452, 633, 100932, 91, 757, 658, 1079, 145, 346, 101440, 102261, 330, 1596, 442, 2128

Offset: 1

N. J. A. Sloane, Apr 08 2016

nonn

Inspired by A266569.
In other words, a(n) = n/2 + a(n/2) if n even, a(n) = 3n+1+a(3n+1) if n odd.
From Seiichi Manyama, Apr 25 2016: (Start)
This sequence was inspired by the Collatz problem (A006577).
The Collatz rule is as follows: If n is even, divide it by 2, otherwise multiply it by 3 and add 1 (A006370).
For example, starting with n = 3, one gets the sequence 3, 10, 5, 16, 8, 4, 2, 1. So a(3) = 10 + 5 + 16 + 8 + 4 + 2 + 1 = 46. (End) [Comment edited by N. J. A. Sloane, Apr 25 2016]

Seiichi Manyama, Table of n, a(n) for n = 1..20000
Seiichi Manyama, Table of n, a(n) for n = 1..650289
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370 (Collatz step), A006577 (trajectory length), A033493 (sum including n).

Cf. A266569, A271473.

A270814 := proc(n)
        local a, traj ;
        a := 0 ;
        traj := n ;
        while traj > 1 do
                if type(traj, 'even') then
                        traj := traj/2 ;
                else
                        traj := 3*traj+1 ;
                end if;
                a := a+traj ;
        end do:
        return a;
end proc:
[seq(A270814(n),n=1..60)];

a(n) = my(ret=n-1); while((n>>=valuation(n,2)) > 1, ret+=5*n+2; n=3*n+1); ret; \\ Kevin Ryde, Dec 10 2022

Typo in definition corrected by Gionata Neri, Apr 08 2016

A049074 Ulam's conjecture (steps to return n to 1 after division by 2 and, if needed, multiplication by 3 with 1 added).

Original entry on oeis.org

8, 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

Enoch Haga

Appeared in School Science and Mathematics in 1982.

			Beginning at n=1, algorithm produces s+t+a=8.
a(3) = 49 because the trajectory of n=3 is (3, 10, 5, 16, 8, 4, 2, 1) and these numbers sum to 49. - _David Radcliffe_, Aug 28 2025

Enoch Haga, Problem, School Science and Mathematics, Nov 1983, vol. 83, no 7, page 628.
LaBar, Problem #3929, School Science and Mathematics, Dec 1982, vol. 82 no 8, page 715.

Almost the same as A033493.

Cf. A049067.

def a(n):
    if n==1: return 8
    l=[n]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        l.append(n)
        if n<2: break
    return sum(l)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

A225748 Numbers n for which the sum of the numbers in the Collatz (3x+1) iteration of n is prime.

Original entry on oeis.org

2, 4, 12, 16, 22, 38, 48, 52, 64, 66, 67, 90, 93, 132, 148, 149, 155, 163, 165, 185, 201, 208, 222, 229, 230, 237, 242, 264, 268, 275, 289, 309, 324, 332, 339, 351, 352, 359, 362, 363, 373, 382, 384, 401, 403, 465, 471, 474, 485, 507, 517, 518, 528, 532

Offset: 1

Jayanta Basu, May 14 2013

nonn

			12 is a member since 12 + 6 + 3 + 10 + 5 + 16 + 8 + 4 + 2 + 1 = 67 is a prime.

Cf. A033493.

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &]; Select[Range[500],PrimeQ[Total[coll[#]]] &]

A347532 a(n) is the sum of the nonpowers of 2 in the 3x+1 sequence that starts at n.

Original entry on oeis.org

0, 0, 18, 0, 5, 24, 257, 0, 308, 15, 228, 36, 88, 271, 663, 0, 183, 326, 488, 35, 21, 250, 602, 60, 627, 114, 101409, 299, 411, 693, 101073, 0, 810, 217, 509, 362, 504, 526, 2313, 75, 101300, 63, 1307, 294, 466, 648, 100948, 108, 775, 677, 1099, 166, 368, 101463, 102285, 355

Offset: 1

Omar E. Pol, Sep 05 2021

nonn

a(n) is the sum of the nonpowers of 2 in the n-th row of A347270.

a(n) = 0 if and only if n is a power of 2.

			For n = 6 the 3x+1 sequence starting at 6 is 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... Only the first four terms are nonpowers of 2. The sum of them is 6 + 3 + 10 + 5 = 24, so a(6) = 24.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A208981 (number of nonpowers of 2).

Cf. A000079, A006370, A033493, A033495, A057716, A135282, A232503, A347270, A347519.

a:= proc(n) option remember; `if`(n=2^ilog2(n), 0,
      n+a(`if`(n::odd, 3*n+1, n/2)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 05 2021

a[n_] := Plus @@ Select[NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, # > 1 &], # != 2^IntegerExponent[#, 2] &]; Array[a, 50] (* Amiram Eldar, Sep 06 2021 *)

From Alois P. Heinz, Sep 05 2021: (Start)
a(n) = A033493(n) - 2 * A232503(n) + 1.
a(n) = A033493(n) - 2^(A135282(n)+1) + 1. (End)

A375268 Row sums of A375266.

Original entry on oeis.org

1, 3, 4, 7, 36, 9, 288, 15, 13, 46, 259, 19, 119, 302, 51, 31, 214, 27, 519, 66, 309, 281, 633, 39, 658, 145, 40, 330, 442, 76, 101104, 63, 292, 248, 540, 55, 535, 557, 158, 106, 101331, 344, 1338, 325, 96, 679, 100979, 79, 806, 708, 265, 197, 399, 81, 102316, 386

Offset: 1

Paolo Xausa, Aug 09 2024

Cf. A375265, A375266, A375267, A375280.

Cf. A033493, A285098.

A375265[n_] := Which[Divisible[n, 3], n/3, Divisible[n, 2], n/2, True, 3*n + 1];
Array[Total[NestWhileList[A375265, #, # > 1 &]] &, 100]

a(n) = Sum_{k = 1..A375267(n) + 1} A375266(n,k).

A375910 Row sums of A350279.

Original entry on oeis.org

1, 4, 9, 48, 13, 41, 61, 24, 30, 72, 69, 151, 86, 40, 53, 538, 74, 128, 109, 100, 110, 182, 69, 507, 135, 81, 93, 395, 129, 217, 599, 132, 139, 249, 220, 460, 182, 161, 177, 850, 121, 340, 267, 140, 158, 448, 631, 1625, 232, 173, 182, 708, 233, 389, 504, 220, 242, 428

Offset: 1

Paolo Xausa, Sep 02 2024

1

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A349407, A350279, A375909, A375911.

Cf. A033493, A285098.

FarkasStep[x_] := Which[Divisible[x, 3], x/3, Mod[x, 4] == 3, (3*x + 1)/2, True, (x + 1)/2];
Array[Total[FixedPointList[FarkasStep, 2*# - 1]] - 1 &, 100]

a(n) = Sum_{k = 1..A375909(n) + 1} A350279(n,k).

A225866 Numbers n such that the sum of the numbers in the Collatz (3x+1) iteration of n is a perfect square.

Original entry on oeis.org

1, 3, 5, 33, 60, 245, 304, 372, 1265, 1568, 1756, 1799, 1856, 2409, 2532, 2976, 3100, 3281, 3376, 3394, 3813, 5637, 5972, 6147, 6538, 7213, 7299, 7896, 7966, 8371, 10419, 11526, 13411, 13856, 14168, 15024, 15283, 15709, 16506, 16577, 16916, 19212, 19829, 21372

Offset: 1

Michel Lagneau, May 18 2013

nonn

			60 is in the sequence because 60 + 30 + 15 + 46 + 23 + 70 + 35 + 106 + 53 + 160 + 80 + 40 + 20 + 10 + 5 + 16 + 8 + 4 + 2 + 1 = 784 = 28^2.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A033493.

collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]; Select[Range[22000], IntegerQ[Sqrt[Total[collatz[#]]]]&]

A374843 Number of indices i in [n] such that in the trajectory of i for the Collatz (3x+1) problem the sum and the number of terms are coprime or the trajectory is not finite.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 21, 22, 22, 23, 23, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 38, 39, 40, 40, 40, 40, 40, 40, 40, 40, 40, 41, 42, 43, 44, 45

Offset: 1

Alois P. Heinz, Jul 22 2024

nonn

Terms in the trajectories for the Collatz (3x+1) problem can be used to approximate the value of Pi. This method was found by Roland Yéléhada (see the links below).

			a(5) = 4 = 1 + 1 + 1 + 1 + 0, because gcd(1,1) = gcd(2,3) = gcd(8,49) = gcd(3,7) = 1 and gcd(6,36) > 1.
a(1000) = 606 -> sqrt(6*1000/a(1000)) = 3.14658387763... .

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Manon Bischoff, Pi is everywhere, even in the Collatz problem (in German), Spektrum der Wissenschaft, 2022
Jean-Paul Delahaye, The number Pi is everywhere (in French), interstices, 2017
Alois P. Heinz, Plot of sqrt(6*n/a(n)) - Pi for n = 1..1000000
Wikipedia, Basel Problem
Wikipedia, Collatz Conjecture
Wikipedia, Iverson bracket
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to the number Pi

Cf. A000796, A003989, A006370, A006577, A008908, A013661, A033493, A059956.

b:= proc(n) option remember; [n, 1]+
     `if`(n=1, 0, b(`if`(n::even, n/2, 3*n+1)))
    end:
a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+1-signum(igcd(b(n)[])-1))
    end:
seq(a(n), n=1..68);

b[n_] := b[n] = {n, 1} + If[n == 1, {0, 0}, b[If[EvenQ[n], n/2, 3*n + 1]]];
a[n_] := a[n] = If[n < 1, 0, a[n - 1] + 1 - Sign[GCD @@ b[n] - 1]];
Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Sep 04 2024, after Alois P. Heinz *)

a(n) = a(n-1) + [gcd(A008908(n), A033493(n)) = 1] for n >= 1 with a(0) = 0, where [] is the Iverson bracket.
Limit_{n->oo} sqrt(6*n/a(n)) = Pi = A000796.
Limit_{n->oo} a(n)/n = A059956.
Limit_{n->oo} n/a(n) = A013661.

A386437 Alternating sum of the numbers in the trajectory of n for the 3x+1 problem.

Original entry on oeis.org

1, 1, -13, 3, -6, 19, -24, 5, 19, 16, -9, -7, -23, 38, -212, 11, 14, -1, -85, 4, -22, 31, -181, 31, 72, 49, -21488, -10, -46, 242, 21412, 21, -89, 20, -134, 37, -9, 123, -104, 36, -21433, 64, -104, 13, -43, 227, 21475, 17, -16, -22, -246, 3, -63, 21542, 21040, 66, 75, 104

Offset: 1

Luca Santarsiero, Jul 21 2025

Conjecture: Let P(n) be the set of distinct positive prime numbers that appear in the first n terms of the sequence. Let Q(n) be the subset of those primes that appear with frequency >= 2. Then, as n -> oo, the ratio |Q(n)|/|P(n)| tends to 11/100. Verified for n <= 5 * 10^7.
a(n) < A033493(n), for all n > 1.
Conjecture: Let S(n) be the set of positive prime numbers that appear in the first n terms of the sequence. Then, as n -> oo, the ratio |S(n)|/n tends to 0. Verified for n <= 10^6.

			a(3) = 3 - 10 + 5 - 16 + 8 - 4 + 2 - 1 = -13.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A033493, A006577, A070165.

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..58);  # Alois P. Heinz, Jul 24 2025

a[n_] := Module[{x = n, seq = {n}},While[x != 1, x = If[EvenQ[x], x/2, 3 x + 1]; AppendTo[seq, x]];Total[seq*Table[(-1)^i, {i, 0, Length[seq] - 1}]]]Table[a[n], {n, 1, 60}]

a(n) = my(s=n, m=n, k=-1); while (m != 1, if (m%2, m=3*m+1, m=m/2); s+=k*m; k=-k); s; \\ Michel Marcus, Jul 25 2025

a(n) = Sum_{k=1..A006577(n)} A070165(n,k) * (-1)^(k+1).

A274243 Numbers n for which the sum of the odd numbers in the Collatz (3x+1) iteration of n is prime.

Original entry on oeis.org

11, 13, 22, 26, 44, 52, 53, 67, 88, 104, 105, 106, 113, 121, 131, 134, 165, 176, 187, 208, 210, 211, 212, 226, 227, 231, 242, 243, 257, 261, 262, 268, 273, 289, 291, 293, 325, 329, 330, 352, 373, 374, 416, 419, 420, 422, 424, 431, 447, 452, 454, 461, 462, 473

Offset: 1

Michel Lagneau, Jul 01 2016

nonn

The corresponding primes are 47, 19, 47, 19, 47, 19, 59, 263, 47, 19, 947, 59, 199, 19777, 419, 263, 20359, 47, 1759, 19, 947, 1291, 59, 199, 569, 23813, 19777, 20173,...

			11 is in the sequence because the Collatz trajectory of 11 is 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the sum of the odd terms is 11 + 17 + 13 + 5 + 1 = 47 is prime.

Cf. A033493, A225748.

lst={};coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&];a:=Select[coll[n],OddQ[#]&];Do[s=Sum[a[[i]],{i,1,Length[a]}];If[PrimeQ[s],AppendTo[lst,n]],{n,1,500}];lst

cp's OEIS Frontend

A270814 a(1)=0; thereafter a(2k)=k+a(k), a(2k+1)=6k+4+a(6k+4).

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A049074 Ulam's conjecture (steps to return n to 1 after division by 2 and, if needed, multiplication by 3 with 1 added).

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A225748 Numbers n for which the sum of the numbers in the Collatz (3x+1) iteration of n is prime.

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Mathematica

A347532 a(n) is the sum of the nonpowers of 2 in the 3x+1 sequence that starts at n.

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A375268 Row sums of A375266.

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A375910 Row sums of A350279.

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A225866 Numbers n such that the sum of the numbers in the Collatz (3x+1) iteration of n is a perfect square.

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Mathematica

A374843 Number of indices i in [n] such that in the trajectory of i for the Collatz (3x+1) problem the sum and the number of terms are coprime or the trajectory is not finite.

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