A372444 -id:A372444 - cp's OEIS frontend

A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

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).

A372452 Number of terms of A086893 in the interval [A372444(n), A372444(1+n)].

Original entry on oeis.org

2, 6, 10, 21, 41, 80, 162, 324, 646, 1294, 2586, 5173, 10345, 20691, 41381, 82760, 165522, 331044, 662089, 1324177, 2648353, 5296707, 10593413, 21186827, 42373652, 84747305, 169494609, 338989216, 677978435, 1355956869, 2711913736, 5423827472, 10847654948, 21695309896, 43390619791, 86781239586, 173562479173, 347124958344

Offset: 0

Antti Karttunen, May 05 2024

nonn

The formula involving A372451 and A372453 shows that each term is at most +-1 from the corresponding term of A372451, that are the first differences of A372449.

			Between A372444(0)=27 and A372444(1)=165 there are two terms (53 and 85) of A086893, therefore a(0) = 2.
Between A372444(1)=165 and A372444(2)=8021 there are six terms (213, 341, 853, 1365, 3413, 5461) of A086893, therefore a(1) = 6.
Between A372444(2)=8021 and A372444(3)=12408149 there are 10 terms (13653, 21845, 54613, 87381, 218453, 349525, 873813, 1398101, 3495253, 5592405) of A086893, therefore a(2) = 10.

Antti Karttunen, Table of n, a(n) for n = 0..1001

Column 14 of A372285.

Cf. A086893, A372286, A372444, A372449, A372451, A372453.

A372452(n) = A372451(n)+(A372453(n)<=0)-(A372453(1+n)<0); \\ Uses also code from A372451 and A372453.

a(n) = A372286(A372444(n)).

a(n) = A372451(n) + [A372453(n)<=0] - [A372453(1+n)<0], where [ ] is the Iverson bracket, yielding 1 or 0 depending on whether the given inequivalence holds or does not hold.

A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

Original entry on oeis.org

6, -48, 2560, -1572864, -3848290697216, 6649092007880460460883968, -18999521285301737936647902825311679255527123058688, 76895533293152762966220781422103876125697362804839499718093497881599910128103059800826635129716736

Offset: 0

Antti Karttunen, May 05 2024

sign

The difference between A372444(n) and the term of A086893 with the same binary length.

			The term of A086893 that has same binary length as A372444(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 A372444(1) = 165 is 213, therefore a(1) = 165-213 = -48.

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

Cf. A000523, A086893, A372444, A372448, A372449, A372453.

A000523(n) = logint(n,2);
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); };
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372454(n) = { my(x=A372444(n)); (x - A086893(1+A000523(x))); };

A372454(n) = if(!n, A372453(n), (4^A372448(n-1))*A372453(n));

a(n) = A372444(n) - A086893(1+A000523(A372444(n))).

a(0) = A372453(0) = 6; and for n > 0, a(n) = 4^A372448(n-1) * A372453(n).

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];

A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

Original entry on oeis.org

27, 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395, 593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429, 911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 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

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

Cf. A000265, A075677, A371093.
Column 14 of A372283, Row 13 of A256598 (but only up to the first 1).
Row 1 of A372560.
From term 47 to the first 1 same as A088593.
Sequences derived from this one or related to:
  A372444,
  A372445 column index of a(n) in array A257852,
  A372362 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372447 binary lengths minus 1,
  A372446 a(n) xored with the term of A086893 having the same binary length,
  A372453 a(n) minus the term of A086893 having the same binary length.

R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372443(n) = { my(x=27); while(n, x=R(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = R(a(n-1)), where R(n) = (3*n+1)/2^A371093(n) = A000265(3*n+1).

For n > 0, a(n) = R(A372444(n-1)) = A000265(1+3*A372444(n-1)).

A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

Original entry on oeis.org

1, 4, 9, 19, 39, 79, 160, 322, 645, 1292, 2585, 5171, 10344, 20689, 41379, 82759, 165520, 331043, 662087, 1324175, 2648352, 5296705, 10593412, 21186825, 42373651, 84747303, 169494607, 338989215, 677978433, 1355956867, 2711913735, 5423827471, 10847654946, 21695309894, 43390619790, 86781239584, 173562479171, 347124958343

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, A372449.

A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));

a(n) = A371093(A372444(n)).

a(0) = 1, and for n > 0, a(n) = 2*a(n-1) + A371093(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)).

A372451 a(n) = A372449(1+n) - A372449(n).

Original entry on oeis.org

3, 5, 11, 21, 40, 81, 161, 324, 647, 1293, 2587, 5172, 10346, 20691, 41380, 82761, 165521, 331045, 662088, 1324177, 2648354, 5296706, 10593414, 21186826, 42373653, 84747305, 169494608, 338989217, 677978434, 1355956869, 2711913737, 5423827472, 10847654948, 21695309895, 43390619792, 86781239585, 173562479173, 347124958345

Offset: 0

Antti Karttunen, May 05 2024

nonn

a(n) tells how many bits the length of the binary expansion grows when we go from A372444(n) to A372444(1+n).

Antti Karttunen, Table of n, a(n) for n = 0..1001

First differences of A372449.
Column 14 of A372356.
Cf. A000523, A372443, A372444.

A372451(n) = (A372449(1+n)-A372449(n)); \\ Needs also code from A372449.

a(n) = A000523(A372444(1+n)) - A000523(A372444(n)).

cp's OEIS Frontend

A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

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A372452 Number of terms of A086893 in the interval [A372444(n), A372444(1+n)].

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A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

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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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A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

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A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with 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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A372445 a(n) = A371092(A372443(n)).

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A372451 a(n) = A372449(1+n) - A372449(n).

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