A371094 -id:A371094 - cp's OEIS frontend

A372292 Numbers that occur more than once in the odd bisection of A371094.

21, 117, 213, 309, 341, 405, 501, 597, 693, 789, 885, 981, 1077, 1173, 1269, 1365, 1461, 1557, 1653, 1749, 1845, 1877, 1941, 2037, 2133, 2229, 2325, 2421, 2517, 2613, 2709, 2805, 2901, 2997, 3093, 3189, 3285, 3381, 3413, 3477, 3573, 3669, 3765, 3861, 3957, 4053, 4149, 4245, 4341, 4437, 4533, 4629, 4725, 4821, 4917

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur more than once in array A371100.

			21 is present because A371094(1) = A371094(3) = 21.
87381 is present because A371094(85) = A371094(213) = A371094(7281) = A371094(14563) = 87381.
185685 is present because A371094(469) = A371094(15473) = A371094(30947) = 185685.

Antti Karttunen, Table of n, a(n) for n = 1..15532

Setwise difference A372290 \ A372291.

Cf. A144864 (subsequence after its initial 1), A371094, A371100.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372292(n) = if(!(n%2),0,my(c=0); forstep(k=1,n,2,if(A371094(k)==n,c++)); (c>1));

search_up_to = 1398101;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372292list(up_to_n) = { my(v=vector((1+up_to_n)/2), x, lista=List([])); forstep(k=1,up_to_n,2,x=A371094(k); if(x <= up_to_n, v[(x+1)/2]++)); for(i=1,(1+up_to_n)/2,if(v[i]>1, listput(lista,i+i-1))); Vec(lista); };
v372292 = A372292list(search_up_to);
A372292(n) = v372292[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];

A372293 Odd numbers that do not occur in the odd bisection of A371094.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Odd numbers that do not occur in array A371100.

Antti Karttunen, Table of n, a(n) for n = 1..14930

Setwise difference A005408 \ A372290.
Cf. A371094, A371100.
Subsequences: A004767, A017077.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372293(n) = if(!(n%2),0,forstep(k=1,n,2,if(A371094(k)==n,return(0))); (1));

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.

Original entry on oeis.org

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

A371100 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) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

Original entry on oeis.org

21, 21, 45, 341, 117, 69, 341, 725, 213, 93, 5461, 1877, 1109, 309, 117, 5461, 11605, 3413, 1493, 405, 141, 87381, 30037, 17749, 4949, 1877, 501, 165, 87381, 185685, 54613, 23893, 6485, 2261, 597, 189, 1398101, 480597, 283989, 79189, 30037, 8021, 2645, 693, 213, 1398101, 2970965, 873813, 382293, 103765, 36181, 9557, 3029, 789, 237

Offset: 1

Antti Karttunen and Ali Sada, Apr 18 2024

			The top left corner of the array:
n\k|      1       2       3        4        5        6        7        8
---+--------------------------------------------------------------------------
1  |     21,     45,     69,      93,     117,     141,     165,     189, ...
2  |     21,    117,    213,     309,     405,     501,     597,     693, ...
3  |    341,    725,   1109,    1493,    1877,    2261,    2645,    3029, ...
4  |    341,   1877,   3413,    4949,    6485,    8021,    9557,   11093, ...
5  |   5461,  11605,  17749,   23893,   30037,   36181,   42325,   48469, ...
6  |   5461,  30037,  54613,   79189,  103765,  128341,  152917,  177493, ...
7  |  87381, 185685, 283989,  382293,  480597,  578901,  677205,  775509, ...
8  |  87381, 480597, 873813, 1267029, 1660245, 2053461, 2446677, 2839893, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Cf. A007283, A079319, A257852, A371094, A371101.
Cf. A372351 (same terms, in different order), A372290 (sorted into ascending order, without duplicates), A372293 (odd numbers that do not occur here).
Leftmost column is A144864 duplicated, without its initial 1.
Row 1: A102603.

A371100[n_, k_] := 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
Table[A371100[n - k + 1, k], {n, 10}, {k, n}] (* Paolo Xausa, Apr 21 2024 *)

up_to = 55;
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371100list(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] = A371100sq((a-(col-1)),col))); (v); };
v371100 = A371100list(up_to);
A371100(n) = v371100[n];

A(n, k) = A007283(n)*A257852(n,k) + A079319(n).
A(n, k) = A371094(A257852(n,k)).
A(n+2, k) = 5 + 16*A(n,k).

A372287 Array read by upward antidiagonals: A(n, k) = A371092(A372283(n, k)), n,k >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 28 2024

A(n, k) gives the column index of A372282(n, k) [or equally, of A372283(n, k)] in array A257852.

Collatz conjecture is equal to the claim that in each column 1 will eventually appear.

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13  14 15  16 17 18 19  20
---+---------------------------------------------------------------
1  | 1, 1, 1, 2, 2, 3, 1, 4, 3, 5, 1, 6, 4,  7, 2,  8, 5, 9, 2, 10,
2  | 1, 1, 1, 3, 2, 3, 1, 6, 1, 2, 1, 9, 5,  6, 3, 12, 4, 1, 2, 15,
3  | 1, 1, 1, 3, 3, 1, 1, 9, 1, 3, 1, 1, 2,  8, 3, 18, 5, 1, 3, 12,
4  | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 12, 1, 27, 2, 1, 3, 17,
5  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 18, 1, 21, 3, 1, 1,  4,
6  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 16, 3, 1, 1,  5,
7  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 1, 23, 1, 1, 1,  2,
8  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 18, 1, 1, 1,  3,
9  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 26, 1, 1, 1,  3,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 1, 39, 1, 1, 1,  1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 30, 1, 1, 1,  1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 44, 1, 1, 1,  1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 30, 1, 66, 1, 1, 1,  1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 44, 1, 99, 1, 1, 1,  1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 66, 1, 75, 1, 1, 1,  1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 99, 1, 28, 1, 1, 1,  1,
17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 75, 1, 42, 1, 1, 1,  1,
18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 63, 1, 1, 1,  1,
19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 48, 1, 1, 1,  1,
20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 63, 1, 71, 1, 1, 1,  1,

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

Cf. A257852, A371092, A371094, A372282, A372283, A372288.

Cf. also A371097 (array giving every fourth column, 1, 5, 9, ...), A371103 (array giving every even numbered column), also array A371101.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
A372287sq(n,k) = A371092(A372283sq(n,k));
A372287list(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] = A372287sq((a-(col-1)),col))); (v); };
v372287 = A372287list(up_to);
A372287(n) = v372287[n];

A(n, k) = A371092(A372282(n,k)) = A371092(A372283(n,k)).

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];

A372354 Array read by upward antidiagonals: A(n, k) = A000523(A372282(n, k)), n,k >= 1, where A000523(x) is one less than the number of bits in the binary expansion of x.

Original entry on oeis.org

0, 4, 1, 12, 4, 2, 28, 12, 8, 2, 60, 28, 20, 5, 3, 124, 60, 44, 10, 6, 3, 252, 124, 92, 19, 13, 6, 3, 508, 252, 188, 40, 26, 11, 8, 3, 1020, 508, 380, 84, 51, 24, 20, 6, 4, 2044, 1020, 764, 172, 104, 52, 44, 11, 7, 4, 4092, 2044, 1532, 348, 212, 108, 92, 19, 16, 6, 4, 8188, 4092, 3068, 700, 428, 220, 188, 40, 36, 13, 12, 4

Offset: 1

Antti Karttunen, Apr 30 2024

			Array begins:
n\k|    1     2     3    4    5    6     7    8     9   10    11   12   13   14
---+-----------------------------------------------------------------------------
1  |    0,    1,    2,   2,   3,   3,    3,   3,    4,   4,    4,   4,   4,   4,
2  |    4,    4,    8,   5,   6,   6,    8,   6,    7,   6,   12,   7,   8,   7,
3  |   12,   12,   20,  10,  13,  11,   20,  11,   16,  13,   28,  11,  14,  12,
4  |   28,   28,   44,  19,  26,  24,   44,  19,   36,  26,   60,  24,  29,  23,
5  |   60,   60,   92,  40,  51,  52,   92,  40,   76,  51,  124,  52,  58,  44,
6  |  124,  124,  188,  84, 104, 108,  188,  84,  156, 104,  252, 108, 115,  84,
7  |  252,  252,  380, 172, 212, 220,  380, 172,  316, 212,  508, 220, 232, 165,
8  |  508,  508,  764, 348, 428, 444,  764, 348,  636, 428, 1020, 444, 468, 326,
9  | 1020, 1020, 1532, 700, 860, 892, 1532, 700, 1276, 860, 2044, 892, 940, 650,

Cf. A000523, A371094, A372282, A372356 (columnwise first differences), A372357.

Row 1 is 0 followed by A113473.

up_to = 78;
A000523(n) = logint(n,2);
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)));
A372354sq(n,k) = A000523(A372282sq(n,k));
A372354list(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] = A372354sq((a-(col-1)),col))); (v); };
v372354 = A372354list(up_to);
A372354(n) = v372354[n];

A371103 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) = A371092(A371102(n, k)), n,k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 3, 3, 4, 1, 1, 1, 6, 5, 1, 1, 1, 9, 2, 6, 1, 1, 1, 1, 3, 9, 7, 1, 1, 1, 1, 3, 1, 6, 8, 1, 1, 1, 1, 1, 1, 8, 12, 9, 1, 1, 1, 1, 1, 1, 12, 18, 1, 10, 1, 1, 1, 1, 1, 1, 18, 27, 1, 15, 11, 1, 1, 1, 1, 1, 1, 27, 21, 1, 12, 9, 12, 1, 1, 1, 1, 1, 1, 21, 16, 1, 17, 7, 18, 13, 1, 1, 1, 1, 1, 1, 16, 23, 1, 4, 2, 27, 5, 14

Offset: 1

Antti Karttunen, Apr 21 2024

A(n, k) gives the column index of A371102(n, k) in array A257852.

			Array begins:
n\k| 1  2   3   ...
---+--------------------------------------------------------------------
1  | 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
2  | 1, 3,  3,  6,  2,  9,  6, 12,  1, 15,  9, 18,  5, 21, 12, 24,  4,
3  | 1, 3,  1,  9,  3,  1,  8, 18,  1, 12,  7, 27,  2,  8, 17, 36,  5,
4  | 1, 1,  1,  1,  3,  1, 12, 27,  1, 17,  2, 21,  3, 12,  4, 54,  2,
5  | 1, 1,  1,  1,  1,  1, 18, 21,  1,  4,  2, 16,  3, 18,  5, 81,  3,
6  | 1, 1,  1,  1,  1,  1, 27, 16,  1,  5,  3, 23,  1, 27,  2, 16,  3,
7  | 1, 1,  1,  1,  1,  1, 21, 23,  1,  2,  3, 18,  1, 21,  3, 23,  1,
8  | 1, 1,  1,  1,  1,  1, 16, 18,  1,  3,  1, 26,  1, 16,  3, 18,  1,
9  | 1, 1,  1,  1,  1,  1, 23, 26,  1,  3,  1, 39,  1, 23,  1, 26,  1,
10 | 1, 1,  1,  1,  1,  1, 18, 39,  1,  1,  1, 30,  1, 18,  1, 39,  1,
11 | 1, 1,  1,  1,  1,  1, 26, 30,  1,  1,  1, 44,  1, 26,  1, 30,  1,
12 | 1, 1,  1,  1,  1,  1, 39, 44,  1,  1,  1, 66,  1, 39,  1, 44,  1,
13 | 1, 1,  1,  1,  1,  1, 30, 66,  1,  1,  1, 99,  1, 30,  1, 66,  1,
14 | 1, 1,  1,  1,  1,  1, 44, 99,  1,  1,  1, 75,  1, 44,  1, 99,  1,
15 | 1, 1,  1,  1,  1,  1, 66, 75,  1,  1,  1, 28,  1, 66,  1, 75,  1,
16 | 1, 1,  1,  1,  1,  1, 99, 28,  1,  1,  1, 42,  1, 99,  1, 28,  1,
17 | 1, 1,  1,  1,  1,  1, 75, 42,  1,  1,  1, 63,  1, 75,  1, 42,  1,
18 | 1, 1,  1,  1,  1,  1, 28, 63,  1,  1,  1, 48,  1, 28,  1, 63,  1,
19 | 1, 1,  1,  1,  1,  1, 42, 48,  1,  1,  1, 71,  1, 42,  1, 48,  1,
20 | 1, 1,  1,  1,  1,  1, 63, 71,  1,  1,  1, 54,  1, 63,  1, 71,  1,
21 | 1, 1,  1,  1,  1,  1, 48, 54,  1,  1,  1, 80,  1, 48,  1, 54,  1,

Cf. A000027 (row 1), A257852, A371092, A371102.

Cf. also arrays A371097, A371101.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371102sq(n,k) = if(1==n,4*k-1,A371094(A371102sq(n-1,k)));
A371103sq(n,k) = A371092(A371102sq(n,k));
A371103list(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] = A371103sq((a-(col-1)),col))); (v); };
v371103 = A371103list(up_to);
A371103(n) = v371103[n];

A372285 Array read by upward antidiagonals: A(n,k) is the number of terms of A086893 in the interval [b(n, k), b(n+1, k)], n,k >= 1, where b = A372282.

Original entry on oeis.org

5, 9, 4, 17, 9, 7, 33, 17, 13, 2, 65, 33, 25, 5, 4, 129, 65, 49, 10, 6, 3, 257, 129, 97, 22, 13, 6, 6, 513, 257, 193, 45, 26, 14, 13, 3, 1025, 513, 385, 89, 54, 29, 25, 4, 4, 2049, 1025, 769, 177, 109, 57, 49, 9, 10, 3, 4097, 2049, 1537, 353, 217, 113, 97, 22, 21, 6, 9, 8193, 4097, 3073, 705, 433, 225, 193, 45, 41, 13, 17, 2

Offset: 1

Antti Karttunen, Apr 27 2024

			Array begins:
n\k|    1     2      3     4     5     6      7     8      9    10     11
---+----------------------------------------------------------------------
1  |    5,    4,     7,    2,    4,    3,     6,    3,     4,    3,     9,
2  |    9,    9,    13,    5,    6,    6,    13,    4,    10,    6,    17,
3  |   17,   17,    25,   10,   13,   14,    25,    9,    21,   13,    33,
4  |   33,   33,    49,   22,   26,   29,    49,   22,    41,   26,    65,
5  |   65,   65,    97,   45,   54,   57,    97,   45,    81,   54,   129,
6  |  129,  129,   193,   89,  109,  113,   193,   89,   161,  109,   257,
7  |  257,  257,   385,  177,  217,  225,   385,  177,   321,  217,   513,
8  |  513,  513,   769,  353,  433,  449,   769,  353,   641,  433,  1025,
9  | 1025, 1025,  1537,  705,  865,  897,  1537,  705,  1281,  865,  2049,
10 | 2049, 2049,  3073, 1409, 1729, 1793,  3073, 1409,  2561, 1729,  4097,
11 | 4097, 4097,  6145, 2817, 3457, 3585,  6145, 2817,  5121, 3457,  8193,
12 | 8193, 8193, 12289, 5633, 6913, 7169, 12289, 5633, 10241, 6913, 16385,
etc.
The count includes also the starting and/or ending point, if either of them is a term of A086893. For example, when going from A372282(2,1) = 21 to A372282(3,1) = 5461, we count terms A086893(5..13) = [21, 53, 85, 213, 341, 853, 1365, 3413, 5461], nine in total, therefore A(2,1) = 9.
When going from A371102(1,8) = 15 to A371102(2,8) = 93, we count terms 21, 53, 85 of A086893 in the interval [15, 93], therefore A(1,8) = 3.

Cf. A086893, A372282, A372286.

up_to = 78;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
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)));
A372286(n) = { my(u=A371094(n), k1); for(i=1,oo,if(A086893(i)>=n,k1=i-1; break)); for(i=k1,oo,if(A086893(i)>u,return(i-k1-1))); };
A372285sq(n,k) = A372286(A372282sq(n,k));
A372285list(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] = A372285sq((a-(col-1)),col))); (v); };
v372285 = A372285list(up_to);
A372285(n) = v372285[n];

A(n, k) = A372286(A372282(n, k)).

cp's OEIS Frontend

A372292 Numbers that occur more than once in the odd bisection of A371094.

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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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A372293 Odd numbers that do not occur in the odd bisection of A371094.

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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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A371100 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) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

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A372287 Array read by upward antidiagonals: A(n, k) = A371092(A372283(n, k)), n,k >= 1.

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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".

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A372354 Array read by upward antidiagonals: A(n, k) = A000523(A372282(n, k)), n,k >= 1, where A000523(x) is one less than the number of bits in the binary expansion of x.

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A371100 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) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.