A371093 -id:A371093 - cp's OEIS frontend

A075677 Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.

1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53, 7, 59, 31, 65, 17, 71, 37, 77, 5, 83, 43, 89, 23, 95, 49, 101, 13, 107, 55, 113, 29, 119, 61, 125, 1, 131, 67, 137, 35, 143, 73, 149, 19, 155, 79, 161, 41, 167, 85, 173, 11, 179, 91, 185, 47, 191, 97, 197

Offset: 1

T. D. Noe, Sep 25 2002

The even-indexed terms a(2i+2) = 6i+5 = A016969(i), i >= 0 [Comment corrected by Bob Selcoe, Apr 06 2015]. The odd-indexed terms are the same as A067745. Note that this sequence is A016789 with all factors of 2 removed from each term. Also note that a(4i-1) = a(i). No multiple of 3 is in this sequence. See A075680 for the number of iterations of R required to yield 1.
From Bob Selcoe, Apr 06 2015: (Start)
All numbers in this sequence appear infinitely often.
From Eq. 1 and Eq. 2 in Formulas: Eq. 1 is used with 1/3 of the numbers in this sequence, Eq. 2 is used with 2/3 of the numbers.
(End)
Empirical: For arbitrary m, Sum_{n=2..A007583(m)} (a(n) - a(n-1)) = 0. - Fred Daniel Kline, Nov 23 2015
From Wolfdieter Lang, Dec 07 2021: (Start)
Only positive numbers congruent to 1 or 5 modulo 6 appear.
i) For the sequence entry with value A016921(m), for m >= 0, that is, a value from {1, 7, 13, ...}, the indices n are given by the row of array A178415(2*m+1, k), for k >= 1.
ii) For the sequence entry with value A007528(m), for m >= 1, that is, a value from {5, 11, 17, ...}, the indices n are given by the row of array A178415(2*m, k), for k >= 1.
See also the array A347834 with permuted row numbers and columns k >= 0. (End)

			a(11) = 1 because 21 is the 11th odd number and R(21) = 64/64 = 1.
From _Wolfdieter Lang_, Dec 07 2021: (Start)
i) 1 (mod 6) entry 1 = A016921(0) appears for n = A178415(1, k) = A347834(1, k-1) (the arrays), for k >= 1, that is, for {1, 5, 21, ..} = A002450.
ii) 5 (mod 6) entry 11 = A007528(2) appears for n = A178415(4, k) = A347835(3, k-1) (the arrays), for k >= 1, that is, for {7, 29, 117, ..} = A072261. (End)

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E16, pp. 330-336.
Victor Klee and Stan Wagon, Old and new unsolved problems in plane geometry and number theory, The Mathematical Association of America, 1991, p. 225, C(2n+1) = a(n+1), n >= 0.
Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see p. 57, also (90-9), p. 306.

T. D. Noe, Table of n, a(n) for n = 1..1000
Marc Chamberland, Una actualizacio del problema 3x + 1, Butl. Soc. Catalana Mat. (18), 19-45, 2003.
Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A000265, A000302, A002450, A007528, A016789, A016921, A016969, A065677, A072197, A072261, A075680, A081294, A178415, A191669, A329480, A347834.
Cf. A006370, A014682 (for non-reduced Collatz maps), A087230 (A371093), A371094.
Odd bisection of A139391.
Even bisection of A067745, which is also the odd bisection of this sequence.
After the initial 1, the second leftmost column of A256598.
Row 2 of A372283.

a075677 = a000265 . subtract 2 . (* 6) -- Reinhard Zumkeller, Jan 08 2014

f:=proc(n) local t1;
if n=1 then RETURN(1) else
t1:=3*n+1;
while t1 mod 2 = 0 do t1:=t1/2; od;
RETURN(t1); fi;
end;
# N. J. A. Sloane, Jan 21 2011

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]
v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; Table[f[2*n - 1], {n, 66}] (* L. Edson Jeffery, May 06 2015 *)

a(n)=n+=2*n-1;n>>valuation(n,2) \\ Charles R Greathouse IV, Jul 05 2013

from sympy import divisors
def a(n):
    return max(d for d in divisors(n) if d % 2)
print([a(6*n - 2) for n in range(1, 101)]) # Indranil Ghosh, Apr 15 2017, after formula by Reinhard Zumkeller

a(n) = A000265(6*n-2) = A000265(3*n-1). - Reinhard Zumkeller, Jan 08 2014
From Bob Selcoe, Apr 05 2015: (Start)
For all n>=1 and for every k, there exists j>=0 dependent upon n and k such that either:
Eq. 1: a(n) = (3n-1)/2^(2j+1) when k = ((4^(j+1)-1)/3) mod 2^(2j+3). Alternatively: a(n) = A016789(n-1)/A081294(j+1) when k = A002450(j+1) mod A081294(j+2). Example: n=51; k=101 == 5 mod 32, j=1. a(51) = 152/8 = 19.
  or
Eq. 2: a(n) = (3n-1)/4^j when k = (5*2^(2j+1) - 1)/3 mod 4^(j+1). Alternatively: a(n) = A016789(n-1)/A000302(j) when k = A072197(j) mod A000302(j+1). Example: n=91; k=181 == 53 mod 64, j=2. a(91) = 272/16 = 17.
(End) [Definition corrected by William S. Hilton, Jul 29 2017]
a(n) = a(n + g*2^r) - 6*g, n > -g*2^r. Examples: n=59; a(59)=11, r=5. g=-1: 11 = a(27) = 5 - (-1)*6; g=1: 11 = a(91) = 17 - 1*6; g=2: 11 = a(123) = 23 - 2*6; g=3: 11 = a(155) = 29 - 3*6; etc. - Bob Selcoe, Apr 06 2015
a(n) = a((1 + (3*n - 1)*4^(k-1))/3), k>=1 (cf. A191669). - L. Edson Jeffery, Oct 05 2015
a(n) = a(4n-1). - Bob Selcoe, Aug 03 2017
a(n) = A139391(2n-1). - Antti Karttunen, May 06 2024
Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Aug 26 2024
G.f.: Sum_{k>=1} ((3 + 2*(-1)^k)*x^(3*2^(k - 1) - (-2)^k/3 + 1/3) + (3 - 2*(-1)^k)*x^(2^(k - 1) - (-2)^k/3 + 1/3))/(x^(2^k) - 1)^2. - Miles Wilson, Oct 26 2024

A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

Original entry on oeis.org

1, 21, 7, 21, 13, 341, 19, 45, 25, 117, 31, 69, 37, 341, 43, 93, 49, 213, 55, 117, 61, 5461, 67, 141, 73, 309, 79, 165, 85, 725, 91, 189, 97, 405, 103, 213, 109, 1877, 115, 237, 121, 501, 127, 261, 133, 1109, 139, 285, 145, 597, 151, 309, 157, 5461, 163, 333, 169, 693, 175, 357, 181, 1493, 187, 381, 193, 789, 199

Offset: 0

Antti Karttunen (proposed by Ali Sada), Apr 19 2024

Construction: take the binary expansion of 3n+1 (A016777(n)), and substitute "01" for all trailing 0-bits that follow after its odd part (= A067745(1+n)), of which there are A371093(n) in total. See the examples.

			For n=1, 3*n+1 = 4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(1) = 21.
For n=6, 3*6+1 = 19, "10011" in binary, and there are no trailing 0's, and no changes, therefore a(6) = 19.
For n=7, 3*7+1 = 22, "10110" in binary, with one trailing 0, which when replaced with 01 gives us 45, "101101" in binary, therefore a(7) = 45.
For n=229, there are e=4 trailing bit expansions 0 -> 01,
  3n+1 = binary  101011  0 0 0 0
  a(n) = binary  101011 01010101

Antti Karttunen, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Plot binary expansion of a(n) arranged with smallest bit at bottom and largest at top, n increasing from left to right for n = 0..3071. Black indicates 1, white indicates 0.
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to binary expansion of n

Cf. A016777, A067745, A075677, A086893, A096773, A372289, A371093.
Cf. A016921, A372351 (even and odd bisection), A372290 (numbers occurring in the latter).
Cf. arrays A372282, A372283 and A371096, A371100, A371102.
Cf. also A302338.

Array[#2*(2^#3) + ((4^#3) - 1)/3 & @@ {#1, #2, IntegerExponent[#2, 2]} & @@ {#, 3 #1 + 1} &, 67, 0] (* Michael De Vlieger, Apr 19 2024 *)

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };

def A371094(n): return ((m:=3*n+1)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A372289(A016777(n)).

a(2n) = A016777(2n) = A016921(n).

A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n(6k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.

Original entry on oeis.org

3, 1, 7, 13, 9, 11, 5, 29, 17, 15, 53, 37, 45, 25, 19, 21, 117, 69, 61, 33, 23, 213, 149, 181, 101, 77, 41, 27, 85, 469, 277, 245, 133, 93, 49, 31, 853, 597, 725, 405, 309, 165, 109, 57, 35, 341, 1877, 1109, 981, 533, 373, 197, 125, 65, 39

Offset: 1

L. Edson Jeffery, Jul 12 2015

Sequence is a permutation of the odd natural numbers.

Let N_1 denote the set of odd natural numbers, and let |y|_2 denote 2-adic valuation of y. Define the map F : N_1 -> N_1 by F(x) = (3*x + 1)/2^|3*x+1|_2 (cf. A075677). Then row n of A is the set of all x in N_1 for which |3*x + 1|_2 = A371093(x) = n. Hence F(A(n,k)) = 6*k - 3 - 2*(-1)^n.

			From _Ruud H.G. van Tol_, Oct 17 2023, corrected and extended by _Antti Karttunen_, Apr 18 2024: (Start)
Array A begins:
n\k|   1|   2|   3|   4|   5|   6|   7|   8| ...
---+---------------------------------------------
1  |   3,   7,  11,  15,  19,  23,  27,  31, ...
2  |   1,   9,  17,  25,  33,  41,  49,  57, ...
3  |  13,  29,  45,  61,  77,  93, 109, 125, ...
4  |   5,  37,  69, 101, 133, 165, 197, 229, ...
5  |  53, 117, 181, 245, 309, 373, 437, 501, ...
6  |  21, 149, 277, 405, 533, 661, 789, 917, ...
... (End)

Cf. A006370, A075677, A096773 (after its initial 0, column 1 of this array).
Cf. A004767, A017077, A082285, A238477 (rows 1-4).
Cf. A371092, A371093 (column and row indices for odd numbers).
Cf. also arrays A371095, A371096, A371097, A371100, A371101, A371102, A371103.

(* Array: *)
Grid[Table[(2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(2^(n - k + 1)*(6*k - 3 - 2*(-1)^(n - k + 1)) - 1)/ 3, {n, 10}, {k, n}]]

up_to = 105;
A257852sq(n,k) = ((2^n * (6*k - 3 - 2*(-1)^n) - 1)/3);
A257852list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A257852sq((a-(col-1)),col))); (v); };
v257852 = A257852list(up_to);
A257852(n) = v257852[n]; \\ Antti Karttunen, Apr 18 2024

From Ruud H.G. van Tol, Oct 17 2023: (Start)
A(n,k+1) = A(n,k) + 2^(n+1).
A(n+2,k) = A(n,k)*4 + 1.
A(1,k) = A004767(k-1).
A(2,k) = A017077(k-1).
A(3,k) = A082285(k-1).
A(4,k) = A238477(k). (End)
For all odd positive numbers n, A(A371093(n), A371092(n)) = n. - Antti Karttunen, Apr 24 2024

A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

Original entry on oeis.org

27, 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395, 593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429, 911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, May 01 2024

nonn

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

Cf. A000265, A075677, A371093.
Column 14 of A372283, Row 13 of A256598 (but only up to the first 1).
Row 1 of A372560.
From term 47 to the first 1 same as A088593.
Sequences derived from this one or related to:
  A372444,
  A372445 column index of a(n) in array A257852,
  A372362 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372447 binary lengths minus 1,
  A372446 a(n) xored with the term of A086893 having the same binary length,
  A372453 a(n) minus the term of A086893 having the same binary length.

R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372443(n) = { my(x=27); while(n, x=R(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = R(a(n-1)), where R(n) = (3*n+1)/2^A371093(n) = A000265(3*n+1).

For n > 0, a(n) = R(A372444(n-1)) = A000265(1+3*A372444(n-1)).

A372283 Array read by upward antidiagonals: A(n, k) = R(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1, where Reduced Collatz function R(n) gives the odd part of 3n+1.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 1, 1, 7, 1, 1, 1, 11, 9, 1, 1, 1, 17, 7, 11, 1, 1, 1, 13, 11, 17, 13, 1, 1, 1, 5, 17, 13, 5, 15, 1, 1, 1, 1, 13, 5, 1, 23, 17, 1, 1, 1, 1, 5, 1, 1, 35, 13, 19, 1, 1, 1, 1, 1, 1, 1, 53, 5, 29, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 11, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 35, 25

Offset: 1

Antti Karttunen, Apr 28 2024

Collatz conjecture is equal to the claim that in each column 1 will eventually appear. See also arrays A372287 and A372288.

			Array begins:
n\k| 1  2  3   4   5   6   7   8   9  10  11  12  13   14  15   16  17  18
---+-----------------------------------------------------------------------
1  | 1, 3, 5,  7,  9, 11, 13, 15, 17, 19, 21, 23, 25,  27, 29,  31, 33, 35,
2  | 1, 5, 1, 11,  7, 17,  5, 23, 13, 29,  1, 35, 19,  41, 11,  47, 25, 53,
3  | 1, 1, 1, 17, 11, 13,  1, 35,  5, 11,  1, 53, 29,  31, 17,  71, 19,  5,
4  | 1, 1, 1, 13, 17,  5,  1, 53,  1, 17,  1,  5, 11,  47, 13, 107, 29,  1,
5  | 1, 1, 1,  5, 13,  1,  1,  5,  1, 13,  1,  1, 17,  71,  5, 161, 11,  1,
6  | 1, 1, 1,  1,  5,  1,  1,  1,  1,  5,  1,  1, 13, 107,  1, 121, 17,  1,
7  | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  5, 161,  1,  91, 13,  1,
8  | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 121,  1, 137,  5,  1,
9  | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  91,  1, 103,  1,  1,
10 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 137,  1, 155,  1,  1,
11 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 103,  1, 233,  1,  1,
12 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 155,  1, 175,  1,  1,
13 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 233,  1, 263,  1,  1,
14 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 175,  1, 395,  1,  1,
15 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 263,  1, 593,  1,  1,
16 | 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 395,  1, 445,  1,  1,

Cf. A005408 (row 1), A075677 (row 2), A372443 (column 14).
Cf. A258145 (A075680), A371093.
Arrays derived from this one or related to:
  A372282,
  A372287 the column index of A(n, k) in array A257852,
  A372361 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even,
  A372360 binary weights of A372361.
Cf. also array A371095 (giving every fourth column, 1, 5, 9, ...) and irregular array A256598 which gives the terms of each column, but only down to the first 1.

With[{dmax = 15}, Table[#[[k, n-k+1]], {n, dmax}, {k, n}] & [Array[NestList[(3*# + 1)/2^IntegerExponent[3*# + 1, 2] &, 2*# - 1, dmax - #] &, dmax]]] (* Paolo Xausa, Apr 29 2024 *)

up_to = 91;
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
A372283list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372283sq((a-(col-1)),col))); (v); };
v372283 = A372283list(up_to);
A372283(n) = v372283[n];

For n > 1, A(n, k) = R(A372282(n-1, k)), where R(n) = (3*n+1)/2^A371093(n).

For all k >= 1, A(A258145(k-1), k) = 1 [which is the topmost 1 in each column].

A371092 a(n) = floor((A000265(3*n+1)+5)/6), where A000265 gives the odd part of its argument.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 19 2024

nonn

When a(n) is applied to square array A257852 we obtain square array A002260, or in other words, a(n) applied to any odd number gives the index of the column where it is located in array A257852.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A000265, A002260, A007814, A257852, A371101.

Cf. also A371093.

with(padic): A37109 := n -> floor(1/6*(3*n + 1)/2^ordp(3*n + 1, 2) + 5/6); seq(A37109(n), n = 0 .. 89); # Miles Wilson, Oct 10 2024

A371092[n_] := With[{k = 3*n + 1}, Floor[(k/2^IntegerExponent[k, 2] + 5)/6]];
Array[A371092, 100, 0] (* Paolo Xausa, Apr 20 2024 *)

A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);

a(n) = a(4*n + 1) = a(16*n + 5) = a(64*n + 21) = ... = a(4^k * n + ((4^k)-1)/3).

A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

Original entry on oeis.org

4, 7, 12, 23, 44, 84, 165, 326, 650, 1297, 2590, 5177, 10349, 20695, 41386, 82766, 165527, 331048, 662093, 1324181, 2648358, 5296712, 10593418, 21186832, 42373658, 84747311, 169494616, 338989224, 677978441, 1355956875, 2711913744, 5423827481, 10847654953, 21695309901, 43390619796, 86781239588, 173562479173, 347124958346

Offset: 0

Antti Karttunen, May 04 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A371093, A371094, A372443, A372444, A372447, A372448.

Column 14 of A372354.

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372447(n) = A000523(A372443(n));
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));
A372449(n) = if(0==n,A372447(n),A372447(n)+2*A372448(n-1));

a(n) = A000523(A372444(n)).

a(0) = A372447(0) = 4, and for n > 0, a(n) = A372447(n) + 2*A372448(n-1).

A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

Original entry on oeis.org

1, 4, 9, 19, 39, 79, 160, 322, 645, 1292, 2585, 5171, 10344, 20689, 41379, 82759, 165520, 331043, 662087, 1324175, 2648352, 5296705, 10593412, 21186825, 42373651, 84747303, 169494607, 338989215, 677978433, 1355956867, 2711913735, 5423827471, 10847654946, 21695309894, 43390619790, 86781239584, 173562479171, 347124958343

Offset: 0

Antti Karttunen, May 04 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A371093, A371094, A372443, A372444, A372447, A372449.

A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));

a(n) = A371093(A372444(n)).

a(0) = 1, and for n > 0, a(n) = 2*a(n-1) + A371093(A372443(n)).

A372362 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with Reduced Collatz-function R}.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Antti Karttunen, May 06 2024

nonn

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

Cf. A371093, A372443, A372447, A372448.

The first 41 terms form the row 13 of A351122.

A000265(n) = (n>>valuation(n,2));
A371093(n) = valuation(1+3*n,2);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372362(n) = A371093(A372443(n));

a(n) = A371093(A372443(n)).

A375961 2-adic valuation of 6*n + 2.

Original entry on oeis.org

1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 9, 1

Offset: 0

Ruud H.G. van Tol, Sep 04 2024

6*i+2 is the first (3*x+1)/2 successor of 4*i+1, with i >= 0.

The first occurrence of odd t is before that of t-1.

			a(21) = A007814(6*21+2) = 7.

Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A007814, A016933, A087229, A087230, A096773, A371093, A375782.

a[n_] := IntegerExponent[6*n + 2, 2]; Array[a, 100, 0] (* Amiram Eldar, Sep 04 2024 *)

PARI
```
a(n) = valuation(6*n+2, 2);
```

def A375961(n): return (~(3*n+1)&3*n).bit_length()+1 # Chai Wah Wu, Sep 27 2024

a(n) = A007814(6*n + 2).
a(n) = A371093(n) + 1.
a(n) = A087229(n) - 1.
a(n) = k for n == A096773(k) (mod 2^k), k >= 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Oct 01 2024

cp's OEIS Frontend

A075677 Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.

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A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

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A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.

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A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

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A372283 Array read by upward antidiagonals: A(n, k) = R(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1, where Reduced Collatz function R(n) gives the odd part of 3n+1.

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A371092 a(n) = floor((A000265(3*n+1)+5)/6), where A000265 gives the odd part of its argument.

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A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

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A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n(6k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.