A109812 -id:A109812 - cp's OEIS frontend

A352204 Indices of records in A109812.

1, 2, 3, 5, 7, 8, 11, 15, 18, 21, 23, 27, 29, 32, 38, 40, 50, 53, 56, 59, 70, 75, 84, 89, 101, 103, 106, 108, 113, 156, 161, 163, 164, 188, 239, 293, 307, 328, 417, 420, 425, 431, 433, 480, 491, 546, 617, 631, 633, 648, 649, 651, 890, 892, 1225, 1235, 1245, 1251, 1305, 1307, 1315, 1323, 1324, 1476, 1485, 1517, 1598, 1891, 1898, 1913, 1970

Offset: 1

N. J. A. Sloane and Chai Wah Wu, Mar 28 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..962 (terms 1..113 from N. J. A. Sloane, terms 114..380 terms from Michael De Vlieger)
Rémy Sigrist, C++ program.

Cf. A109812, A352203.

c[] = 0; a[1] = c[1] = r = 1; u = 2; {1}~Join~Reap[Do[k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]]; If[k > r, Sow[i]; Set[r, k]], {i, 2, 1980}]][[-1, -1]] (* _Michael De Vlieger, Mar 29 2022 *)

A352203 Records in A109812.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 18, 32, 33, 36, 40, 48, 64, 66, 72, 96, 128, 129, 130, 132, 136, 192, 193, 194, 196, 256, 257, 320, 322, 328, 384, 448, 512, 528, 640, 768, 770, 800, 1024, 1025, 1028, 1029, 1280, 1281, 1312, 1536, 1538, 1544, 1792, 1793, 2048, 2304, 2560, 2688, 3072, 3073, 3074, 3136, 3200, 3328, 3584, 3592, 4096, 4100, 4101, 4104, 4160, 5120

Offset: 1

N. J. A. Sloane and Chai Wah Wu, Mar 28 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..962 (terms 1..113 from N. J. A. Sloane, terms 114..380 from Michael De Vlieger)
Rémy Sigrist, C++ program.

Cf. A109812, A352204.

c[] = 0; a[1] = c[1] = r = 1; u = 2; Reap[Do[k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]]; If[k > r, Sow[r]; Set[r, k]], {i, 2, 1980}]][[-1, -1]] (* _Michael De Vlieger, Mar 29 2022 *)

A113233 Inverse permutation to A109812.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 16, 5, 10, 7, 20, 12, 26, 14, 28, 8, 13, 11, 22, 19, 24, 30, 39, 17, 33, 36, 41, 44, 51, 54, 76, 15, 18, 25, 43, 21, 37, 34, 48, 23, 31, 60, 80, 46, 57, 85, 104, 27, 66, 62, 71, 64, 90, 83, 109, 68, 88, 100, 102, 74, 112, 107, 162, 29, 35, 32, 45, 42, 61, 67

Offset: 1

Reinhard Zumkeller, Oct 19 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A109812, A113232, A113234, A305370.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a113233 = (+ 1) . fromJust . (`elemIndex` a109812_list)
-- Reinhard Zumkeller, Sep 15 2014

A113234(n) = a(a(n)).

a(63) corrected by Reinhard Zumkeller, Sep 15 2014

A305370 Index of 2^n in A109812.

Original entry on oeis.org

1, 2, 3, 5, 8, 15, 29, 50, 103, 164, 417, 649, 1324, 2684, 5348, 10226, 21298, 41538, 84652, 165034, 322262, 646648, 1283556, 2584860, 5163368, 10271638, 20566708, 40308350, 80758553, 161522919, 322519008, 637135132

Offset: 0

N. J. A. Sloane, Jun 02 2018

It would be nice to have a formula for a(n).

The data suggests that a(n) is growing very roughly like 2^n/3.261. - N. J. A. Sloane, Mar 29 2022

			A109812(15) = 32 = 2^5, so a(5) = 15.

Rémy Sigrist, C++ program for A305370

Cf. A109812, A113233.

a(14)-a(22) from Chai Wah Wu, Jun 04 2018
a(23)-a(24) from Giovanni Resta, Jun 06 2018
a(25)-a(26) from Rémy Sigrist, Dec 05 2021
a(27)-a(28) from Walter Trump, Apr 13 2022
a(29)-a(31) from Walter Trump, Apr 26 2022

A352336 Define a sequence B = {b(i): i >= 1} by b(i) = smallest unused number when A109812(i) is being calculated, and then remove duplicates from B.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 22, 23, 27, 28, 29, 30, 31, 43, 46, 47, 55, 61, 63, 87, 91, 93, 94, 95, 123, 125, 126, 127, 189, 191, 222, 223, 235, 237, 238, 239, 247, 251, 254, 255, 319, 373, 375, 379, 381, 383, 431, 439, 443, 446, 447, 475, 479, 495, 499, 503, 506, 507, 509, 511, 765, 767, 895, 959, 989, 991, 1007, 1023, 1503, 1519, 1531, 1535, 1783

Offset: 1

Michael De Vlieger, Mar 29 2022

Might be called the local minima in A109812.

Also indices of records in A113233. These are the numbers that are the slowest to appear in A109812. They arrive late in A109812 because of having few zeros in their binary expansion. Every number of the form 2^k - 1 is necessarily a member, since any number less than 2^k - 1 must occur earlier in A109812. - David Broadhurst, Aug 17 2022

			The initial terms of A109812 and the smallest missing numbers (smn):
  n a(n) smn
  1  1   2
  2  2   3
  3  4   3
  4  3   5
  5  8   5
  6  5   6
  7  10  6
  8  16  6
  9   6  7
  10  9  7
  11  18 7
  12  12 7
  ...
so the distinct smallest missing numbers are 1, 2, 3, 5, 6, 7, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..221
David Broadhurst, Table of n, A352336(n), A352359(n) for n = 1..221

Cf. A109812, A113233, A352203, A352359.

c[_] = 0; a[1] = c[1] = 1; u = 2; {1}~Join~Reap[Do[k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; If[a[i - 1] == u, Sow[u]; While[c[u] > 0, u++]]; Set[{a[i], c[k]}, {k, i}], {i, 2, nn}]][[-1, -1]]

Edited by N. J. A. Sloane, Apr 26 2022 and May 03 2024

A352359 Indices of local minima in A109812.

Original entry on oeis.org

1, 2, 4, 6, 9, 16, 20, 26, 28, 30, 39, 41, 44, 51, 54, 76, 80, 85, 104, 109, 112, 162, 165, 175, 179, 197, 240, 294, 308, 323, 418, 423, 434, 481, 490, 500, 510, 536, 547, 632, 634, 647, 650, 699, 767, 782, 805, 883, 891, 914, 952, 996, 1009, 1226, 1269, 1271, 1314, 1316, 1322, 1325, 1477, 1483, 1892, 1901, 2419, 2588, 2685, 2803, 2936, 2988, 3555, 3570, 3833

Offset: 1

Michael De Vlieger, Mar 29 2022

These are the indices (in A109812) of numbers that are the slowest to appear in A109812.

Records in A113233.

David Broadhurst, Table of n, a(n) for n = 1..221
David Broadhurst, Table of n, A352336(n), A352359(n) for n = 1..221

Cf. A109812, A113233, A352204, A352336.

c[_] = 0; a[1] = c[1] = 1; u = 2; {1}~Join~Reap[Do[k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; If[a[i - 1] == u, Sow[i - 1]; While[c[u] > 0, u++]]; Set[{a[i], c[k]}, {k, i}], {i, 2, 1300}]][[-1, -1]]

Edited by N. J. A. Sloane, Apr 26 2022

A352575 A109812(n) in binary.

Original entry on oeis.org

1, 10, 100, 11, 1000, 101, 1010, 10000, 110, 1001, 10010, 1100, 10001, 1110, 100000, 111, 11000, 100001, 10100, 1011, 100100, 10011, 101000, 10101, 100010, 1101, 110000, 1111, 1000000, 10110, 101001, 1000010, 11001, 100110, 1000001, 11010, 100101, 1001000, 10111, 1100000, 11011, 1000100, 100011, 11100, 1000011, 101100, 1010000, 100111, 1011000, 10000000

Offset: 1

N. J. A. Sloane, Apr 04 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Bitmap of a(n), n = 1..2^14, horizontally exaggerating each row by 256X.
N. J. A. Sloane, Right-justified table of n, a(n) for n = 1..20000 [Note this is not a b-file.]
N. J. A. Sloane, Right-justified table of n, a(n) for n = 1..10^6 [gzipped file, copied from A352575] [This file was corrupted, but today I replaced it with a corrected version. - _N. J. A. Sloane_, Apr 11 2022]

Cf. A109812, A352576, A352884.

nn = 50, c[] = 0; a[1] = c[1] = 1; u = 2; Do[If[a[i - 1] == u, While[c[u] > 0, u++]]; k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; Set[{a[i], c[k]}, {k, i}], {i, Length[s] + 1, nn}]; Array[FromDigits@ IntegerDigits[a[#], 2] &, nn] (* _Michael De Vlieger, Apr 05 2022 *)

A352917 Indices k where A109812(k)/k reaches a new high point.

Original entry on oeis.org

1, 3, 5, 8, 15, 29, 40, 50, 108, 164, 649, 651, 5509386, 11271059, 45010096

Offset: 1

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 21 2022)

The corresponding values of A109812(k) are given in A352918.
This is a subset of A352204.
The slow growth of A109812(k)/k (see Examples section) suggests that A109812(k)/k is bounded. That is, it appears there is a constant c (between 3.7 and 4) such that A109812(k) < c*k for all k.

			Let c(k) denote A109812(k). The first 15 record high-points of c(k)/k are as follows:
[c(k)/k, k, c(k), "binary(c(n))"]
[1.000000000, 1, 1, "1"]
[1.333333333, 3, 4, "100"]
[1.600000000, 5, 8, "1000"]
[2.000000000, 8, 16, "10000"]
[2.133333333, 15, 32, "100000"]
[2.206896552, 29, 64, "1000000"]
[2.400000000, 40, 96, "1100000"]
[2.560000000, 50, 128, "10000000"]
[2.962962963, 108, 320, "101000000"]
[3.121951220, 164, 512, "1000000000"]
[3.155624037, 649, 2048, "100000000000"]
[3.539170507, 651, 2304, "100100000000"]
[3.616182275, 5509386, 19922944, "1001100000000000000000000"]
[3.721304271, 11271059, 41943040, "10100000000000000000000000"]
[3.727433952, 45010096, 167772160, "1010000000000000000000000000"]
The values of k and c(k) form the present sequence and A352918.

Cf. A109812, A352203, A352204, A352918.

A113232 A109812(A109812(n)).

Original entry on oeis.org

1, 2, 3, 4, 16, 8, 9, 7, 5, 6, 33, 12, 24, 14, 66, 10, 21, 25, 11, 18, 26, 20, 96, 36, 38, 17, 39, 32, 52, 19, 27, 49, 34, 72, 74, 13, 37, 76, 40, 100, 48, 56, 65, 15, 70, 28, 43, 23, 57, 104, 64, 112, 135, 22, 139, 120, 67, 137, 92, 68, 71, 128, 131, 98, 60, 88, 136, 130, 51

Offset: 1

Reinhard Zumkeller, Oct 19 2005

nonn

Inverse integer permutation of A113234;

A113233(a(n)) = a(A113233(n)) = A109812(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A109812, A113233, A113234.

a113232 = a109812 . a109812  -- Reinhard Zumkeller, Sep 15 2014

Erroneous data corrected by Reinhard Zumkeller, Sep 15 2014

A305372 A109812(2k+1).

Original entry on oeis.org

1, 4, 8, 10, 6, 18, 17, 32, 24, 20, 36, 40, 34, 48, 64, 41, 25, 65, 37, 23, 27, 35, 67, 80, 88, 29, 129, 97, 45, 132, 69, 73, 74, 70, 71, 51, 131, 192, 160, 144, 84, 54, 46, 134, 194, 138, 162, 146, 139, 133, 196, 256, 208, 62, 55, 258, 322, 83, 89, 264, 161, 145, 152, 148, 135, 259, 163, 90, 266, 154, 106, 288

Offset: 0

N. J. A. Sloane, Jun 02 2018

nonn

It appears that this is the same as A305369(4k+3)/2.

This is an (apparent) link between A109812 and A305369, and a formula or recurrence would benefit both sequences.

N. J. A. Sloane, Table of n, a(n) for n = 0..4999

Cf. A109812, A305369.

The graphs of A109812, A252867, A305369, A305372 all have roughly the same, mysterious, fractal-like structure. - N. J. A. Sloane, Jun 03 2018

cp's OEIS Frontend

A352204 Indices of records in A109812.

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A352203 Records in A109812.

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A113233 Inverse permutation to A109812.

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A305370 Index of 2^n in A109812.

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A352336 Define a sequence B = {b(i): i >= 1} by b(i) = smallest unused number when A109812(i) is being calculated, and then remove duplicates from B.

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A352359 Indices of local minima in A109812.

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Mathematica

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A352575 A109812(n) in binary.

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Mathematica

A352917 Indices k where A109812(k)/k reaches a new high point.

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A113232 A109812(A109812(n)).

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