A371103 -id:A371103 - cp's OEIS frontend

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.

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

A371102 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 4*k-1, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

3, 21, 7, 5461, 45, 11, 357913941, 1109, 69, 15, 1537228672809129301, 873813, 3413, 93, 19, 28356863910078205288614550619314017621, 1466015503701, 22369621, 2261, 117, 23, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 25790417485112089060398421, 6004799503160661, 873813, 11605, 141, 27

Offset: 1

Antti Karttunen, Apr 21 2024

			Array begins:
n\k|         1       2         3       4         5         6         7
---+--------------------------------------------------------------------
1  |         3,      7,       11,     15,       19,       23,       27,
2  |        21,     45,       69,     93,      117,      141,      165,
3  |      5461,   1109,     3413,   2261,    11605,     3413,     8021,
4  | 357913941, 873813, 22369621, 873813, 72701269, 22369621, 12408149,

Cf. A004767 (row 1), A102603 (row 2), A371094.

Cf. also arrays A257852, A371096, A371100, A371103.

up_to = 105;
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)));
A371102list(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] = A371102sq((a-(col-1)),col))); (v); };
v371102 = A371102list(up_to);
A371102(n) = v371102[n];

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)).

A371097 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(A371095(n, k)), n,k >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 21 2024

A(n, k) gives the column index of A371095(n, k) [or equally, of A371096(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, 2, 1, 5, 4,  8, 2, 11, 7, 14,  4, 17, 10, 20,  1,  23, 13,
3  | 1, 3, 1, 2, 5, 12, 2,  9, 2, 21,  6,  4, 14, 30,  1,  18, 10,
4  | 1, 3, 1, 3, 2, 18, 3,  7, 2,  8,  9,  5, 21, 45,  1,  26, 14,
5  | 1, 1, 1, 3, 3, 27, 3,  2, 3, 12,  1,  2,  8, 17,  1,  39, 21,
6  | 1, 1, 1, 1, 3, 21, 1,  2, 3, 18,  1,  3, 12,  4,  1,  30,  8,
7  | 1, 1, 1, 1, 1, 16, 1,  3, 1, 27,  1,  3, 18,  5,  1,  44, 12,
8  | 1, 1, 1, 1, 1, 23, 1,  3, 1, 21,  1,  1, 27,  2,  1,  66, 18,
9  | 1, 1, 1, 1, 1, 18, 1,  1, 1, 16,  1,  1, 21,  3,  1,  99, 27,
10 | 1, 1, 1, 1, 1, 26, 1,  1, 1, 23,  1,  1, 16,  3,  1,  75, 21,
11 | 1, 1, 1, 1, 1, 39, 1,  1, 1, 18,  1,  1, 23,  1,  1,  28, 16,
12 | 1, 1, 1, 1, 1, 30, 1,  1, 1, 26,  1,  1, 18,  1,  1,  42, 23,
13 | 1, 1, 1, 1, 1, 44, 1,  1, 1, 39,  1,  1, 26,  1,  1,  63, 18,
14 | 1, 1, 1, 1, 1, 66, 1,  1, 1, 30,  1,  1, 39,  1,  1,  48, 26,
15 | 1, 1, 1, 1, 1, 99, 1,  1, 1, 44,  1,  1, 30,  1,  1,  71, 39,
16 | 1, 1, 1, 1, 1, 75, 1,  1, 1, 66,  1,  1, 44,  1,  1,  54, 30,
17 | 1, 1, 1, 1, 1, 28, 1,  1, 1, 99,  1,  1, 66,  1,  1,  80, 44,
18 | 1, 1, 1, 1, 1, 42, 1,  1, 1, 75,  1,  1, 99,  1,  1, 120, 66,
19 | 1, 1, 1, 1, 1, 63, 1,  1, 1, 28,  1,  1, 75,  1,  1, 180, 99,
20 | 1, 1, 1, 1, 1, 48, 1,  1, 1, 42,  1,  1, 28,  1,  1, 270, 75,
21 | 1, 1, 1, 1, 1, 71, 1,  1, 1, 63,  1,  1, 42,  1,  1, 405, 28,

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

Cf. A000027 (row 1), A257852, A371092, A371095, A371096.

Cf. also arrays A371101, A371103.

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); };
A371095sq(n,k) = if(1==n,8*k-7,R(A371095sq(n-1,k)));
A371097sq(n,k) = A371092(A371095sq(n,k));
A371097list(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] = A371097sq((a-(col-1)),col))); (v); };
v371097 = A371097list(up_to);
A371097(n) = v371097[n];

A(n, k) = A371092(A371095(n, k)) = A371092(A371096(n, k)).

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)).

cp's OEIS Frontend

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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A371102 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 4*k-1, and A(n+1, k) = A371094(A(n, k)), 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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A371097 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(A371095(n, k)), n,k >= 1.

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

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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.