A371100 -id:A371100 - cp's OEIS frontend

A371101 Array A read by upward antidiagonals in which the entry A(n,k) = A371092(A371100(n, k)), n,k >= 1.

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

Offset: 1

Antti Karttunen, Apr 19 2024

A(n, k) gives the column index of A371100(n, k) in array A257852.

			The 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, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
2  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
3  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
4  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
5  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
6  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
7  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
8  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
...

Cf. A257852, A371092, A371100.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371101sq(n,k) = A371092(A371100sq(n,k));
A371101list(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] = A371101sq((a-(col-1)),col))); (v); };
v371101 = A371101list(up_to);
A371101(n) = v371101[n];

A(n, k) = A371092(A371100(n, k)).

A(n, k) = A(n+2, k).

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

A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

Original entry on oeis.org

1, 21, 3, 5461, 21, 5, 357913941, 5461, 341, 7, 1537228672809129301, 357913941, 1398101, 45, 9, 28356863910078205288614550619314017621, 1537228672809129301, 23456248059221, 1109, 117, 11, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 28356863910078205288614550619314017621, 6602346876188694799461995861, 873813, 11605, 69, 13

Offset: 1

Antti Karttunen, Apr 28 2024

			Array begins:
n\k|    1     2        3     4      5     6        7     8      9     10
---+----------------------------------------------------------------------
1  |    1,    3,       5,    7,     9,   11,      13,   15,    17,    19,
2  |   21,   21,     341,   45,   117,   69,     341,   93,   213,   117,
3  | 5461, 5461, 1398101, 1109, 11605, 3413, 1398101, 2261, 87381, 11605,

Cf. A005408 (row 1), A372351 (row 2, bisection of A371094), A372444 (column 14).
Arrays derived from this one:
  A372283,
  A372285 the number of terms of A086893 in the interval [A(n, k), A(1+n, k)],
  A372287 the column index of A(n, k) in array A257852,
  A372288 the sum of digits of A(n, k) in "Jacobsthal greedy base",
  A372353 differences between A(n,k) and the largest term of A086893 <= A(n,k),
  A372354 floor(log_2(.)) of terms, A372356 (and their columnwise first differences),
  A372359 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.
Cf. also arrays A371096, A371102 that give subsets of columns of this array, and array A371100 that gives the terms of the row 2 in different order.

up_to = 28;
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)));
A372282list(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] = A372282sq((a-(col-1)),col))); (v); };
v372282 = A372282list(up_to);
A372282(n) = v372282[n];

A371096 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) = 8*k-7, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

1, 21, 9, 5461, 117, 17, 357913941, 11605, 213, 25, 1537228672809129301, 72701269, 87381, 309, 33, 28356863910078205288614550619314017621, 3752999689475413, 91625968981, 30037, 405, 41, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 27043212804868893898596335048021, 100743818301219097892181, 760567125, 79189, 501, 49

Offset: 1

Antti Karttunen, Apr 21 2024

			Array begins:
n\k|         1         2            3          4           5          6
---+--------------------------------------------------------------------
1  |         1,        9,          17,        25,         33,        41,
2  |        21,      117,         213,       309,        405,       501,
3  |      5461,    11605,       87381,     30037,      79189,     48469,
4  | 357913941, 72701269, 91625968981, 760567125, 1968526677, 299193685,

Cf. A371094, A017077 (row 1).
Every fourth column (1, 5, 9, 13, 17, ...) of array A372282.
Cf. also arrays A257852, A371100 and A371102.

up_to = 28;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371096sq(n,k) = if(1==n,8*k-7,A371094(A371096sq(n-1,k)));
A371096list(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] = A371096sq((a-(col-1)),col))); (v); };
v371096 = A371096list(up_to);
A371096(n) = v371096[n];

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];

A372290 Numbers that occur in the odd bisection of A371094.

Original entry on oeis.org

21, 45, 69, 93, 117, 141, 165, 189, 213, 237, 261, 285, 309, 333, 341, 357, 381, 405, 429, 453, 477, 501, 525, 549, 573, 597, 621, 645, 669, 693, 717, 725, 741, 765, 789, 813, 837, 861, 885, 909, 933, 957, 981, 1005, 1029, 1053, 1077, 1101, 1109, 1125, 1149, 1173, 1197, 1221, 1245, 1269, 1293, 1317, 1341, 1365, 1389

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur in array A371100.

			21 is present because A371094(1) = A371094(3) = 21.
45 is present because A371094(7) = 45.
87381 is present because A371094(85) = A371094(213) = A371094(7281) = A371094(14563) = 87381.

Antti Karttunen, Table of n, a(n) for n = 1..62122

Union of A372291 and A372292.

Cf. A102603 (subsequence), A371094, A371100.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372290(n) = if(!(n%2),0,forstep(k=1,n,2,if(A371094(k)==n,return(1))); (0));

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372290list(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]>0, listput(lista,i+i-1))); Vec(lista); };

A372351 Odd bisection of A371094.

Original entry on oeis.org

21, 21, 341, 45, 117, 69, 341, 93, 213, 117, 5461, 141, 309, 165, 725, 189, 405, 213, 1877, 237, 501, 261, 1109, 285, 597, 309, 5461, 333, 693, 357, 1493, 381, 789, 405, 3413, 429, 885, 453, 1877, 477, 981, 501, 87381, 525, 1077, 549, 2261, 573, 1173, 597, 4949, 621, 1269, 645, 2645, 669, 1365, 693, 11605, 717

Offset: 1

Antti Karttunen, Apr 28 2024

nonn

Row 2 of A372282.
Cf. A371094, and array A371100 (gives the same terms, in different order).
Cf. A372290 (the range of this sequence), A372291 (numbers that occur only once), A372292 (more than once), A372293 (odd numbers not occurring here).

Table[With[{e = IntegerExponent[6*n - 2, 2]}, (6*n - 2)*2^e + (4^e - 1)/3], {n, 100}] (* Paolo Xausa, Apr 29 2024 *)

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372351(n) = A371094(n+n-1);

def A372351(n): return ((m:=6*n-2)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A371094(2*n-1).

A372291 Numbers that occur exactly once in the odd bisection of A371094.

Original entry on oeis.org

45, 69, 93, 141, 165, 189, 237, 261, 285, 333, 357, 381, 429, 453, 477, 525, 549, 573, 621, 645, 669, 717, 725, 741, 765, 813, 837, 861, 909, 933, 957, 1005, 1029, 1053, 1101, 1109, 1125, 1149, 1197, 1221, 1245, 1293, 1317, 1341, 1389, 1413, 1437, 1485, 1493, 1509, 1533, 1581, 1605, 1629, 1677, 1701, 1725, 1773, 1797

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur exactly once in array A371100.

			45 is present because A371094(k) = 45 for no other odd number than k=7.

Antti Karttunen, Table of n, a(n) for n = 1..25365

Cf. A371094, A371100.

Setwise difference A372290 \ A372292.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372291(n) = if(!(n%2),0,my(c=0); forstep(k=1,n,2,if(A371094(k)==n,c++;if(c>1,return(0)))); (c));

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

Original entry on oeis.org

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];

cp's OEIS Frontend

A371101 Array A read by upward antidiagonals in which the entry A(n,k) = A371092(A371100(n, k)), n,k >= 1.

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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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A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

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A371096 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) = 8*k-7, and A(n+1, k) = A371094(A(n, k)), 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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A372290 Numbers that occur in the odd bisection of A371094.

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PARI

PARI

A372351 Odd bisection of 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.