A372443 -id:A372443 - cp's OEIS frontend

A372447 a(n) = A000523(A372443(n)); One less than the binary length of the n-th iterate of 27 with Reduced Collatz-function R.

4, 5, 4, 5, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 8, 9, 8, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 11, 9, 10, 11, 11, 9, 8, 8, 5, 4, 5, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 03 2024

nonn

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

Cf. A000523, A372443.

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); };
A372447(n) = A000523(A372443(n));

A372446 a(n) = A372358(A372443(n)).

Original entry on oeis.org

14, 28, 10, 26, 18, 62, 116, 44, 14, 92, 50, 78, 60, 122, 82, 222, 260, 232, 114, 46, 44, 78, 252, 106, 138, 410, 354, 774, 1064, 218, 2, 1366, 336, 276, 228, 16, 8, 2, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 03 2024

nonn

These are the bitmasks (or symmetric differences) obtained when the n-th iterate of 27 with Reduced Collatz-function R [= A372443(n), where R(n) = A000265(3*n+1)] is xored with that term of A086893 that has the same binary length. The binary expansions of the terms of A086893 are always of the form 10101...0101 (i.e., alternating 1's and 0's starting and ending with 1) when the binary length is odd, and of the form 110101...0101 (i.e., 1 followed by alternating 1's and 0's, and ending with 1) when n is even. Note that for all n >= 1, R(A086893(2n-1)) = 1, and R(A086893(2n)) = 5 (with R(5) = 1), so the first zero here, a(39) = 0 indicates that the iteration will soon have reached the terminal 1, and indeed, A372443(41) = 1.

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

Column 14 of A372361.

Cf. A000265, A075677, A086893, A372358, A372443.

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372358(n) = bitxor(A086893(1+A000523(n)),n);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372446(n) = A372358(A372443(n));

A372453 a(n) = A372443(n) - A086893(1+A372447(n)).

Original entry on oeis.org

6, -12, 10, -6, -14, 22, -52, 36, 6, -76, 18, -58, 20, -38, -78, 54, -260, 104, -46, 38, 36, -58, 84, -22, 138, -134, -286, 254, -984, 58, 2, -1362, -336, -276, 92, -16, 8, 2, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 05 2024

sign

These are the differences obtained when the term of A086893 that has the same binary length as A372443(n) is subtracted from the latter. Here A372443(n) gives the n-th iterate of 27 with Reduced Collatz-function R, where R(n) = A000265(3*n+1).

Note that for all n >= 1, R(A086893(2n-1)) = 1, and R(A086893(2n)) = 5 (with R(5) = 1), so the first zero here, a(39) = 0 indicates that the iteration will soon have reached the terminal 1, and indeed, A372443(41) = 1.

			The term of A086893 that has same binary length as A372443(0) = 27 is 21 [as 21 = 10101_2 in binary, and 27 = 11011_2 in binary], therefore a(0) = 27-21 = 6.
The term of A086893 that has same binary length as A372443(1) = 41 is 53, therefore a(1) = 41-53 = -12.

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

Cf. A000265, A000523, A086893, A372443.

Cf. also A372446.

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372453(n) = { my(x=A372443(n)); (x - A086893(1+A000523(x))); };

a(n) = A372443(n) - A086893(1+A000523(A372443(n))).

A372560 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = A372443(k-1).

Original entry on oeis.org

27, 165, 41, 8021, 501, 31, 12408149, 48469, 189, 47, 19607957362005, 299193685, 4565, 285, 71, 32439509492992549521282389, 7552911875069269, 1758549, 6869, 429, 107, 58947232705679751034215288252890081792789279233365, 3195535888075328282939605996885, 173230347605, 2643285, 10325, 645, 161

Offset: 1

Antti Karttunen, May 08 2024

			Array begins:
n\k|        1          2        3        4        5         6           7
---+-----------------------------------------------------------------------
1  |       27,        41,      31,      47,      71,      107,        161,
2  |      165,       501,     189,     285,     429,      645,       1941,
3  |     8021,     48469,    4565,    6869,   10325,    31061,     374101,
4  | 12408149, 299193685, 1758549, 2643285, 7951701, 95769941, 9216283989,

Cf. A371094, A372282, A372443 (the top row), A372444 (the leftmost column), A372561.

up_to = 28;
A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372560sq(n,k) = if(1==n,A372443(k-1),A371094(A372560sq(n-1,k)));
A372560list(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] = A372560sq((a-(col-1)),col))); (v); };
v372560 = A372560list(up_to);
A372560(n) = v372560[n];

A372445 a(n) = A371092(A372443(n)).

Original entry on oeis.org

7, 6, 8, 12, 18, 27, 21, 16, 23, 18, 26, 39, 30, 44, 66, 99, 75, 28, 42, 63, 48, 71, 54, 80, 120, 180, 270, 405, 152, 228, 342, 513, 97, 73, 55, 11, 4, 6, 9, 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, 1, 1

Offset: 0

Antti Karttunen, May 01 2024

nonn

a(n) gives the column index of A372443(n), or equally, of A372444(n) in array A257852.

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

Cf. A257852, A371092, A372443, A372444.

Column 14 of A372287, column 7 of A371103.

A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372445(n) = A371092(A372443(n));

a(n) = A371092(A372443(n)) = A371092(A372444(n)).

A256598 Irregular triangle where row n contains the odd terms in the Collatz sequence beginning with 2n+1.

Original entry on oeis.org

1, 3, 5, 1, 5, 1, 7, 11, 17, 13, 5, 1, 9, 7, 11, 17, 13, 5, 1, 11, 17, 13, 5, 1, 13, 5, 1, 15, 23, 35, 53, 5, 1, 17, 13, 5, 1, 19, 29, 11, 17, 13, 5, 1, 21, 1, 23, 35, 53, 5, 1, 25, 19, 29, 11, 17, 13, 5, 1, 27, 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155

Offset: 0

Bob Selcoe, Apr 03 2015

The Collatz function is an integer-valued function given by n/2 if n is even and 3n+1 if n is odd. We build a Collatz sequence by beginning with a natural number and iterating the function indefinitely. It is conjectured that all such sequences terminate at 1.
In this triangle, row n is made up of the odd terms of the Collatz sequence beginning with 2n+1. Therefore, it is conjectured that this sequence is well-defined, i.e., that all rows terminate at 1.
The last index k in row n gives the number of iterations required for the Collatz sequence to terminate if even terms are omitted.
T(n,k)/T(n,k+1) is of form: ceiling(T(n,k)*3/2^j) for some j>=1. Therefore, the coefficients in each row may be read as a series of iterated ceilings, where j may vary. For example, row 3 has initial term 7, which is followed by ceiling(7*3/2), ceiling(ceiling(7*3/2)*3/2), ceiling(ceiling(ceiling(7*3/2)*3/2)*3/4), ceiling(ceiling(ceiling(ceiling(7*3/2)*3/2)*3/4)*3/8), ceiling(ceiling(ceiling(ceiling(ceiling(7*3/2)*3/2)*3/4)*3/8)*3/16).
The length of row n is A258145(n) (set to 0 if 1 is not reached after a finite number of steps). - Wolfdieter Lang, Aug 11 2021

			Triangle starts T(0,0):
n\k   0   1   2   3   4    5   6   7   8   9  10 ...
0:    1
1:    3   5   1
2:    5   1
3:    7   11  17  13  5    1
4:    9   7   11  17  13   5   1
5:    11  17  13  5   1
6:    13  5   1
7:    15  23  35  53  5    1
8:    17  13  5   1
9:    19  29  11  17  13   5   1
10:   21  1
11:   23  35  53  5    1
12:   25  19  29  11  17  13   5   1
...
n=13 starts with 27 and takes 41 steps: (27), 41, 31, 47, 71, 107,... 53, 5, 1, (see A372443).
Row 8 is [17, 13, 5, 1] because it is the subsequence of odd terms for the Collatz sequence starting with 17: [17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1].

Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165, A075677, A258145.
Cf. A372443 (row 13 up to its first 1).
Cf. also array A372283 showing the same terms in different orientation.

f[n_] := NestWhileList[(3*# + 1)/2^IntegerExponent[3*# + 1, 2] &, 2*n + 1, # > 1 &]; Grid[Table[f[n], {n, 0, 12}]] (* L. Edson Jeffery, Apr 25 2015 *)

row(n) = {my(oddn = 2*n+1, vl = List(oddn), x); while (oddn != 1, x = 3*oddn+1; oddn = x >> valuation(x, 2); listput(vl, oddn)); Vec(vl);}
tabf(nn) = {for (n=0, nn, my(rown = row(n)); for (k=1, #rown, print1(rown[k], ", ")); print;);} \\ Michel Marcus, Oct 04 2019

def Collatz(n):
    A = [n]
    b = A[-1]
    while b != 1:
        if is_even(b):
            A.append(b//2)
        else:
            A.append(3*b+1)
    return A
[y for sublist in [[x for x in Collatz(2*n+1) if is_odd(x)] for n in [0..15]] for y in sublist] # Tom Edgar, Apr 04 2015

T(n,0) = 2n+1 and T(n,k) = A000265(3*T(n,k-1)+1) for k>0. - Tom Edgar, Apr 04 2015

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].

A372444 The n-th iterate of 27 with A371094.

Original entry on oeis.org

27, 165, 8021, 12408149, 19607957362005, 32439509492992549521282389, 58947232705679751034215288252890081792789279233365, 259166427025070423330595967015238989905128148712607202753574381749095993394717720069452733214971221

Offset: 0

Antti Karttunen, May 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10

Cf. A371094.
Column 7 of A371102, column 14 of A372282.
Column 1 of A372560.
Sequences derived from this one:
  A372443 obtained when Reduced Collatz-function R is applied to a(n-1), for n > 0,
  A372445 column index of a(n) in array A257852,
  A372448 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372449 binary lengths minus 1; their first differences: A372451,
  A372452 number of terms of A086893 in the interval [a(n), a(1+n)],
  A372454 the difference between a(n) and the term of A086893 with the same binary length.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372444(n) = { my(x=27); while(n, x=A371094(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = A371094(a(n-1)).

A372288 Array read by upward antidiagonals: A(n, k) = A265745(A372282(n, k)), n,k >= 1, where A265745(n) is the sum of digits in "Jacobsthal greedy base".

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 5, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 5, 3

Offset: 1

Antti Karttunen, Apr 28 2024

Collatz conjecture is equal to the claim that each column will eventually settle to constant 1's, somewhere under its topmost row. This works as only the bisection A002450 of Jacobsthal numbers (A001045) contains numbers of the form 4k+1, while the other bisection contains only numbers of the form 4k+3, which do not occur among the range of A372351. See also the comments in A371094.

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13     14 15    16 17 18 19 20    21 22
---+----------------------------------------------------------------------------
1  | 1, 1, 1, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3,     3, 3,    3, 3, 3, 3, 5,    5, 1,
2  | 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 5, 5,     5, 3,    5, 3, 3, 3, 5,    5, 3,
3  | 1, 1, 1, 3, 3, 3, 1, 5, 1, 3, 1, 3, 3,     5, 3,    5, 5, 1, 3, 3,    5, 3,
4  | 1, 1, 1, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3,     5, 3,    3, 3, 1, 3, 5,    5, 3,
5  | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3,     5, 1,    5, 3, 1, 3, 3,    3, 3,
6  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3,     3, 1,    5, 3, 1, 1, 5,    5, 3,
7  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    3, 3, 1, 1, 3,    5, 3,
8  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 3,    3, 3,
9  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     3, 1,    5, 1, 1, 1, 3,    5, 1,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 3,    5, 1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1, 2155, 1, 1, 1, 1,    5, 1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1, 6251, 1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10347, 1,    5, 1, 1, 1, 1,    5, 1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1,    5, 1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    7, 1, 1, 1, 1,    5, 1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1,    7, 1,

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

Cf. A001045, A002450, A265745, A265747, A371094, A372351, A372282, A372283, A372287.

Cf. also array A372561 (formed by columns whose indices in this array are given by A372443).

up_to = 105;
A130249(n) = (#binary(3*n+1)-1);
A001045(n) = (2^n - (-1)^n) / 3;
A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372288sq(n,k) = A265745(A372282sq(n,k));
A372288list(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] = A372288sq((a-(col-1)),col))); (v); };
v372288 = A372288list(up_to);
A372288(n) = v372288[n];

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).

cp's OEIS Frontend

A372447 a(n) = A000523(A372443(n)); One less than the binary length of the n-th iterate of 27 with Reduced Collatz-function R.

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A372446 a(n) = A372358(A372443(n)).

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A372453 a(n) = A372443(n) - A086893(1+A372447(n)).

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A372560 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = A372443(k-1).

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A372445 a(n) = A371092(A372443(n)).

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A256598 Irregular triangle where row n contains the odd terms in the Collatz sequence beginning with 2n+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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A372444 The n-th iterate of 27 with A371094.

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