A372282 -id:A372282 - cp's OEIS frontend

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".

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];

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)).

A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 896, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 56, 4

Offset: 1

Antti Karttunen, Apr 29 2024

Zeros occur in the same locations where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4         5   6  7    8  9        10 11  12                 13
---+---------------------------------------------------------------------------
1  | 0, 0, 0,   2,        4,  6, 0,   2, 4,        6, 0,  2,                 4,
2  | 0, 0, 0,  24,       32, 16, 0,   8, 0,       32, 0, 56,                96,
3  | 0, 0, 0, 256,     6144,  0, 0, 896, 0,     6144, 0,  0,              8192,
4  | 0, 0, 0,   0, 16777216,  0, 0,   0, 0, 16777216, 0,  0,         402653184,
5  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0, 72057594037927936,
6  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0,                 0,

Cf. A086893, A096773, A372352, A372282, A372287.

Cf. also A372285 and A372355 (columnwise first differences).

up_to = 91;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372352(n) = { my(k); for(i=1,oo,k=A086893(i); if(k>n, return(n-A086893(i-1)))); };
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)));
A372353sq(n,k) = A372352(A372282sq(n,k));
A372353list(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] = A372353sq((a-(col-1)),col))); (v); };
v372353 = A372353list(up_to);
A372353(n) = v372353[n];

A372359 Array read by upward antidiagonals: A(n, k) = A372358(A372282(n, k)), n,k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 1408, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 88, 12

Offset: 1

Antti Karttunen, May 01 2024

Zeros occur in the same locations as where they occur in A372353 and where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4     5   6  7     8  9    10 11  12         13             14
---+----------------------------------------------------------------------------
1  | 0, 0, 0,   2,    4,  6, 0,    2, 4,    6, 0,  2,        12,            14,
2  | 0, 0, 0,  24,   32, 16, 0,    8, 0,   32, 0, 88,        96,           112,
3  | 0, 0, 0, 256, 6144,  0, 0, 1408, 0, 6144, 0,  0,      8192,          2560,
4  | 0, 0, 0,   0, 2^24,  0, 0,    0, 0, 2^24, 0,  0, 402653184,       6815744,
5  | 0, 0, 0,   0,    0,  0, 0,    0, 0,    0, 0,  0,      2^56, 4947802324992,
6  | 0, 0, 0,   0,    0,  0, 0,    0, 0,    0, 0,  0,         0,     31 * 2^79,
where 2^56 = 72057594037927936 and 31 * 2^79 = 18738350204026752207945728.

Cf. A003987, A086893, A371094, A372282, A372287, A372354, A372358, A372360 (binary weights).

Cf. also A372353.

up_to = 91;
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372358(n) = bitxor(A086893(1+A000523(n)),n);
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)));
A372359sq(n,k) = A372358(A372282sq(n,k));
A372359list(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] = A372359sq((a-(col-1)),col))); (v); };
v372359 = A372359list(up_to);
A372359(n) = v372359[n];

A(n, k) = A372282(n,k) XOR A086893(1+A372354(n, k)), where XOR is bitwise-xor, A003987.

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).

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)).

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];

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)).

cp's OEIS Frontend

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".

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A372359 Array read by upward antidiagonals: A(n, k) = A372358(A372282(n, k)), n,k >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A372444 The n-th iterate of 27 with A371094.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI