cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A213218 Even numbers in A221715.

Original entry on oeis.org

20, 192, 352, 104, 736, 264, 64, 464, 1088, 640, 1344, 800, 3136, 896, 1952, 2496, 22272, 2752, 1728, 1504, 8576, 4480, 3520, 12672, 7552, 4160, 3840, 11520, 2304, 19712, 9088, 8000, 10880, 14592, 11904, 4864, 21248, 8448, 17664, 26624, 10112, 6528, 5696, 6208, 24704, 22912, 28480, 9536, 39168, 41216, 36736
Offset: 1

Views

Author

N. J. A. Sloane, Mar 02 2013

Keywords

Crossrefs

Cf. A114183, A221715.

A114183 a(1) = 1; for n>1, a(n) = floor(sqrt(a(n-1))) if that number is not already in the sequence, otherwise a(n) = 2a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 5, 10, 3, 6, 12, 24, 48, 96, 9, 18, 36, 72, 144, 288, 576, 1152, 33, 66, 132, 11, 22, 44, 88, 176, 13, 26, 52, 7, 14, 28, 56, 112, 224, 448, 21, 42, 84, 168, 336, 672, 25, 50, 100, 200, 400, 20, 40, 80, 160, 320, 17, 34, 68, 136, 272, 544, 23, 46, 92
Offset: 1

Views

Author

Franklin T. Adams-Watters, Feb 04 2006

Keywords

Comments

One can prove by induction that n must appear in the sequence after [n/2], showing that the sequence is one-to-one; and that frac(log_2(log_2(a(n))) is dense in [0,1), from which it follows that a(n) is onto. - From Franklin T. Adams-Watters, Feb 04 2006
Comment from N. J. A. Sloane, Mar 01 2013: Although the preceding argument seems somewhat incomplete, the result is certainly true: This sequence is a permutation of the natural numbers. Mark Hennings and the United Kingdom Mathematics Trust, and (independently) Max Alekseyev, sent detailed proofs - see the link below.
The sequence consists of a series of "doubling runs", and the starting points and lengths of these runs are in A221715 and A221716 respectively. - N. J. A. Sloane, Jan 27 2013

Links

Crossrefs

Cf. A189419 (inverse), A221715, A221716, A000196, A000523, A213218.
See A222193 and A222194 for records.

Programs

  • Haskell
    a114183 n = a114183_list !! (n-1)
a114183_list = 1 : f [1] where
   f xs@(x:_) = y : f (y : xs) where
     y = if z `notElem` xs then z else 2 * x where z = a000196 x
-- Reinhard Zumkeller, Mar 05 2013
  • Maple
    See A221715.
  • Mathematica
    a[1] = 1; a[n_] := a[n] = With[{an = Floor[Sqrt[a[n-1]]]}, If[FreeQ[Array[a, n-1], an], an, 2*a[n-1]]]; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Apr 23 2013 *)

Extensions

Missing negative in definition inserted by D. S. McNeil, May 26 2010
Entry revised by N. J. A. Sloane, Mar 01 2013

A221716 Length of n-th doubling run in A114183.

Original entry on oeis.org

6, 2, 6, 8, 3, 5, 3, 7, 6, 5, 5, 6, 5, 8, 6, 6, 7, 5, 6, 4, 8, 6, 9, 6, 6, 7, 7, 8, 7, 10, 7, 6, 7, 5, 9, 6, 12, 7, 9, 9, 9, 6, 8, 7, 8, 8, 12, 7, 8, 6, 8, 7, 8, 7, 7, 7, 7, 6, 8, 10, 8, 8, 9, 7, 7, 8, 10, 7, 11, 12, 7, 10, 9, 7, 9, 10, 9, 10, 7, 9, 8, 9, 11, 11, 5, 8, 9, 8, 11, 10, 6, 8, 8, 6, 10, 7, 8, 9, 12, 8, 9, 12, 6, 8, 8, 11, 8, 9, 9, 8, 10, 9, 12
Offset: 1

Views

Author

N. J. A. Sloane, Jan 27 2013

Keywords

Links

Crossrefs

Cf. A114183, A189419, A221715, A213220.

Programs

A213220 Positions in A114183 where doubling runs begin.

Original entry on oeis.org

1, 7, 9, 15, 23, 26, 31, 34, 41, 47, 52, 57, 63, 68, 76, 82, 88, 95, 100, 106, 110, 118, 124, 133, 139, 145, 152, 159, 167, 174, 184, 191, 197, 204, 209, 218, 224, 236, 243, 252, 261, 270, 276, 284, 291, 299, 307, 319, 326, 334, 340, 348, 355, 363, 370, 377, 384, 391, 397, 405, 415, 423, 431, 440, 447, 454, 462, 472
Offset: 1

Views

Author

N. J. A. Sloane, Mar 02 2013

Keywords

Links

Crossrefs

Cf. A114183, A221715, A221716 (first differences).

A213656 Value of A114183 at end of n-th doubling run.

Original entry on oeis.org

32, 10, 96, 1152, 132, 176, 52, 448, 672, 400, 320, 544, 368, 2432, 1568, 1248, 2240, 752, 864, 232, 1920, 1376, 9472, 3104, 1760, 2624, 3264, 7296, 5440, 37376, 12352, 3552, 3776, 976, 7936, 2848, 108544, 21056, 37120, 49152, 56576, 7584, 11136, 6720, 10368, 12928, 231424, 30784, 22400, 4768, 8832, 5952, 9856
Offset: 1

Views

Author

N. J. A. Sloane, Mar 03 2013

Keywords

Links

Crossrefs

Cf. A114183, A221715, A221716, A213220.

A222802 When A114183 decreases in value for the n-th time, dropping to k (say), a(n) is the number of steps earlier that floor(k/2) appeared in A114183.

Original entry on oeis.org

5, 8, 12, 18, 19, 21, 25, 33, 36, 44, 53, 37, 53, 64, 14, 31, 32, 69, 71, 76, 77, 108, 120, 39, 93, 105, 123, 125, 157, 170, 52, 91, 93, 99, 190, 192, 89, 225, 238, 121, 72, 158, 251, 238, 251, 270, 205, 50, 209, 282, 284, 286, 287, 288, 289, 361, 385, 370, 281, 282, 340, 342, 344, 346, 309, 310, 312, 367, 460, 275
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2013

Keywords

Comments

The fact that, when a number k occurs in A114183, floor(k/2) has already appeared, is a key step in the proof that A114183 is a permutation of the natural numbers. This fact is obvious if k is the result of a doubling step. The present sequence is an attempt to gain insight into why it is true when k occurs at a square root step.

Examples

			The first 50 terms of A114183 are:
1, 2, 4, 8, 16, 32, 5, 10, 3, 6, 12, 24, 48, 96, 9, 18, 36, 72, 144, 288, 576, 1152, 33, 66, 132, 11, 22, 44, 88, 176, 13, 26, 52, 7, 14, 28, 56, 112, 224, 448, 21, 42, 84, 168, 336, 672, 25, 50, 100, 200.
The sequence decreases from 32 to 5, from 10 to 3, from 96 to 9, and so on.
The values of k are therefore 5, 3, 9, 33, 11, 13, 7, 21, 25, ...
and the corresponding values of floor(k/2) are 2, 1, 4, 16, 5, 6, 3, 10, 12, ...
Since 2 appeared in A114183 5 steps before 5, a(1) = 5,
since 1 appeared 8 steps before 3, a(2) = 8,
since 4 appeared 12 steps before 9, a(3) = 12, and so on.

Links

Crossrefs

Cf. A114183, A221715, A221716, A213220.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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