A119435 -id:A119435 - cp's OEIS frontend

A119436 Inverse permutation to sequence A119435.

1, 2, 4, 8, 3, 16, 6, 32, 5, 12, 64, 10, 7, 24, 128, 20, 9, 14, 48, 256, 40, 13, 11, 18, 28, 96, 512, 80, 15, 26, 22, 36, 17, 56, 192, 1024, 160, 30, 52, 21, 25, 44, 19, 72, 34, 112, 384, 2048, 29, 320, 23, 60, 27, 104, 42, 50, 88, 38, 144, 68, 31

Offset: 1

Leroy Quet, May 19 2006

Has an unusual graph. - N. J. A. Sloane, May 01 2022

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 500 terms from Sean A. Irvine)
Rémy Sigrist, PARI program

Cf. A119435.

PARI
```
See Links section.
```

More terms from Sean A. Irvine, Mar 25 2013

A353035 a(n) = A119435(2^n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 116, 153, 208, 273, 366, 493, 649, 888, 1161, 1579, 2092, 2784, 3783, 4946, 6772, 8875, 11977, 16065, 21193, 28979, 37823, 51633, 68117, 91045, 123377, 161622, 221441, 289493, 392259, 523328, 692771, 945393

Offset: 0

Michael De Vlieger, Apr 18 2022

Local minima of A119435.
Let U(n,j) be the j-th smallest missing number in A119435(1..n-1). Example: for A119435(1..11), U(12,j) begins {6, 8, 10, 11, 14, ...}. Therefore we may alternatively define A119435(n) = U(n, A030101(n)).
Theorem: A119435(2^k) represents a local minimum. Proof: Observe that A030101(2^k) = 1. 2^k expressed in binary is 1 followed by zeros. When we reverse this number, the leading zeros are trivial and we read the number 1 in the 2^0 place. Therefore we select U(2^k, 1), which is the smallest missing number in A119435(1..n-1). Hence, a(n) = A119435(2^n).
Also positions of 2^n in A119436.

Rémy Sigrist, PARI program

Cf. A030101, A119435, A119436.

a = {1}; nn = 2^14; Do[AppendTo[a, Complement[Range[i + 2 nn], a][[IntegerReverse[i, 2]] ]], {i, 2, nn}]; Array[a[[2^#]] &, Floor@ Log2@ Length@ a - 1, 0]

PARI
```
\\ See Links section.
```

More terms from Rémy Sigrist, Apr 19 2022

A353036 Records in A119435.

Original entry on oeis.org

1, 2, 5, 9, 13, 17, 23, 29, 33, 43, 51, 53, 61, 65, 83, 95, 107, 113, 125, 129, 163, 183, 199, 203, 219, 233, 237, 253, 257, 323, 359, 383, 407, 419, 443, 449, 473, 485, 509, 513, 643, 711, 751, 783, 791, 823, 851, 859, 891, 913, 921, 953, 981, 989, 1021, 1025

Offset: 1

Michael De Vlieger, Apr 18 2022

Terms are odd except for a(2) = 2.
(2^k - 1) +- 2 are terms in this sequence for k > 1.
Let S = A119435. S(2^k + 1) = S(2^k - 1) + 4, while S(2^k) is a local minimum in S.
S(2^k - 1) = 2^(k+1) - 3 and S(2^k + 1) = 2^(k+1) + 1.

Michael De Vlieger, Table of n, a(n) for n = 1..1360
Michael De Vlieger, Bitmap of a(n), n = 1..967, vertical exaggeration 12X.
Michael De Vlieger, Binary expansion of a(n), n = 1..967.

Cf. A030101, A119435, A353035, A353037.

a = {1}; nn = 514; r = 0; Do[AppendTo[a, Complement[Range[i + 2 nn], a][[IntegerReverse[i, 2]] ]], {i, 2, nn}]; Reap[Do[If[# > r, r = #; Sow[r]] &@ a[[i]], {i, Length[a]}]][[-1, -1]]

A353037 Positions of records in A119435.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 31, 33, 35, 39, 47, 55, 63, 65, 67, 71, 79, 87, 95, 111, 119, 127, 129, 131, 135, 143, 159, 175, 191, 207, 223, 239, 255, 257, 259, 263, 271, 287, 303, 319, 351, 367, 383, 415, 431, 447, 479, 495, 511, 513, 515, 519, 527

Offset: 1

Michael De Vlieger, Apr 18 2022

Let S = A119435. S(2^k +- 1) with k > 0 are records in S and thus (2^k - 1) and (2^k + 1) appear in this sequence.

S(2^k - 1) = 2^(k+1) - 3 and S(2^k + 1) = 2^(k+1) + 1, while S(2^k) is a local minimum.

Michael De Vlieger, Table of n, a(n) for n = 1..1360
Michael De Vlieger, Bitmap of a(n), n = 1..967, vertical exaggeration 12X.
Michael De Vlieger, Binary expansion of a(n), n = 1..967.

Cf. A030101, A119435, A353035, A353036.

a = {1}; nn = 527; r = 0; Do[AppendTo[a, Complement[Range[i + 2 nn], a][[IntegerReverse[i, 2]] ]], {i, 2, nn}]; Reap[Do[If[# > r, r = #; Sow[i]] &@ a[[i]], {i, Length[a]}]][[-1, -1]]

A246165 Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 11, 5, 12, 10, 17, 9, 23, 16, 19, 8, 29, 18, 35, 15, 28, 25, 41, 14, 31, 34, 30, 24, 51, 27, 59, 13, 44, 43, 47, 26, 67, 52, 58, 22, 77, 42, 83, 38, 49, 61, 89, 21, 70, 46, 73, 53, 99, 45, 69, 37, 88, 75, 111, 40, 119, 85, 72, 20, 94, 64, 127, 63, 103, 68, 137, 39, 143, 97, 79, 78, 106, 87, 151, 36

Offset: 1

Antti Karttunen, Aug 17 2014

nonn

Terms at a(2^n) are: 1, 2, 3, 5, 8, 13, 20, 32, 48, 71, 105, 156, 236, 354, 542, 815, 1228, ...

Fixed points begin as: 1, 2, 6, 10, 18, 42, 92, 26372, ...

			By definition, a(1) = 1.
After that, for n = 2, when its prime factorization is shifted once right, results A064989(2) = 1, so we select the 1st of still unused positive natural numbers, which is 2, thus a(2) = 2.
For n = 3 = p_2 (3 is the second prime), when its prime factorization is shifted once right, results A064989(3) = 2 = p_1, so we select 2nd of still unused numbers, which is 4, thus a(3) = 4.
For n = 4, like for all powers of two, the result of right shifting is 1, so we select the smallest still unused number, which is 3, thus a(4) = 3.
For n = 5 = p_3, A064989(5) = 3 = p_2, so we select the 3rd smallest still unused number from [5, 6, 7, 8, ...] which is 7, thus a(5) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A246166.
Similar permutations: A119435, A126917.
Cf. A064989.

A266411 a(1) = 1, after which each a(n) = (A004074(n)+1)-th number selected from those not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 7, 5, 10, 12, 14, 13, 16, 15, 11, 9, 18, 20, 22, 24, 23, 26, 28, 27, 30, 29, 25, 32, 31, 21, 19, 17, 34, 36, 38, 40, 42, 41, 44, 46, 48, 47, 50, 52, 51, 54, 53, 49, 56, 58, 57, 60, 59, 55, 62, 61, 45, 43, 64, 63, 39, 37, 35, 33, 66, 68, 70, 72, 74, 76, 75, 78, 80, 82, 84, 83, 86, 88, 90, 89, 92, 94, 93, 96, 95, 91

Offset: 1

Antti Karttunen, Dec 29 2015

nonn

Inverse: A266412.
Cf. A004074.
Similar permutations in Quetian style: A119435, A126917, A246165, A266413.
Cf. also A265901, A265903.

f[n_] := Block[{a = {1}, g, b = Range[2, n]}, g[1] = g[2] = 1; g[x_] := g[x] = g[g[x - 1]] + g[x - g[x - 1]]; Do[{AppendTo[a, #[[1, 1]]], Set[b, Last@ #]} &@ If[# > Length@ b, Break[], TakeDrop[b, {#}]] &@ (2 g[#] - # + 1) &@ k, {k, 2, n}]; a]; f@ 97 (* Michael De Vlieger, Dec 29 2015, Version 10.2, based on Harvey P. Dale at A004074 *)

A266413 a(1) = 1, after which each a(n) = A002487(n)-th number selected from those not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 9, 5, 12, 11, 15, 10, 17, 14, 18, 8, 21, 20, 25, 19, 28, 24, 29, 16, 31, 27, 34, 23, 35, 30, 33, 13, 38, 37, 43, 36, 47, 42, 48, 32, 51, 46, 55, 41, 56, 49, 53, 26, 57, 52, 62, 45, 65, 59, 64, 40, 66, 60, 69, 50, 68, 58, 63, 22, 71, 70, 77, 67, 82, 76, 83, 61, 87, 81, 92, 75, 93, 84, 89, 54, 94, 88, 101, 80

Offset: 1

Antti Karttunen, Dec 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences that are permutations of the natural numbers

Inverse: A266414.
Cf. A002487.
Similar permutations in Quetian style: A119435, A126917, A246165, A266411.
Cf. also A266405.

f[n_] := Block[{a = {1}, g, b = Range[2, n]}, g[1] = 1; g[x_] := g[x] = If[EvenQ@ x, g[x/2], g[(x - 1)/2] + g[(x + 1)/2]]; Do[{AppendTo[a, #[[1, 1]]], Set[b, Last@ #]} &@ If[# > Length@ b, Break[], TakeDrop[b, {#}]] &@ g@ k, {k, 2, n}]; a]; f@ 103 (* Michael De Vlieger, Dec 29 2015, Version 10.2, after N. J. A. Sloane at A002487 *)

A318578 Let k be the greatest odd divisor of n and let S be the sequence of positive integers not in the sequence so far in increasing order. Then a(n) = S(k).

Original entry on oeis.org

1, 2, 5, 3, 9, 7, 13, 4, 17, 12, 21, 10, 25, 18, 29, 6, 33, 23, 37, 16, 41, 28, 45, 14, 49, 34, 53, 24, 57, 39, 61, 8, 65, 44, 69, 31, 73, 50, 77, 22, 81, 55, 85, 38, 89, 60, 93, 19, 97, 66, 101, 46, 105, 71, 109, 32, 113, 76, 117, 52, 121, 82, 125, 11, 129, 87, 133

Offset: 1

Ivan Neretin, Aug 29 2018

nonn

In other words, a(n) = A000265(n)-th positive integer unused so far.
A permutation of the positive integers.
a(n) = 2n - 1 for odd n, a(n) < 2n - 1 otherwise.

			For n=6, the highest odd divisor of n is k = 3. The sequence up to that point is 1, 2, 5, 3, 9. The numbers which are not yet in the sequence (in increasing order) are S = 4, 6, 7, 8, 10, ... and the 3rd of these is 7, which is therefore a(6).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A119435, A101319, A000265.

Fold[Append[#1, Complement[Range[Max[#1] + (r = #2/2^IntegerExponent[#2, 2])], #1][[r]]] &, {1}, Range[2, 67]]

cp's OEIS Frontend

A119436 Inverse permutation to sequence A119435.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A353035 a(n) = A119435(2^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A353036 Records in A119435.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A353037 Positions of records in A119435.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A246165 Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A266411 a(1) = 1, after which each a(n) = (A004074(n)+1)-th number selected from those not yet in the sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A266413 a(1) = 1, after which each a(n) = A002487(n)-th number selected from those not yet in the sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A318578 Let k be the greatest odd divisor of n and let S be the sequence of positive integers not in the sequence so far in increasing order. Then a(n) = S(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica