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.

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A162598 Ordinal transform of A265332.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 2, 1, 5, 6, 7, 3, 8, 4, 2, 1, 9, 10, 11, 12, 5, 13, 14, 6, 15, 7, 3, 16, 8, 4, 2, 1, 17, 18, 19, 20, 21, 9, 22, 23, 24, 10, 25, 26, 11, 27, 12, 5, 28, 29, 13, 30, 14, 6, 31, 15, 7, 3, 32, 16, 8, 4, 2, 1, 33, 34, 35, 36, 37, 38, 17, 39, 40, 41, 42, 18, 43, 44, 45, 19, 46, 47
Offset: 1

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Author

Franklin T. Adams-Watters, Jul 07 2009

Keywords

Comments

This is a fractal sequence.
It appears that each group of 2^k terms starts with 1 and ends with the remaining powers of two from 2^k down to 2^1.
From Antti Karttunen, Jan 09-12 2016: (Start)
This is ordinal transform of A265332, which is modified A051135 (with a(1) = 1, instead of 2). - after Franklin T. Adams-Watters' original definition for this sequence.
A000079 (powers of 2) indeed gives the positions of ones in this sequence. This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 of PDF), which together also imply the pattern observed above, more clearly represented as:
a(2) = 1.
a(3..4) = 2, 1.
a(6..8) = 4, 2, 1.
a(13..16) = 8, 4, 2, 1.
a(28..31) = 16, 8, 4, 2, 1.
etc.
(End)

Links

Crossrefs

Cf. A004001, A051135, A080677, A087686.
Row index of A265901, column index of A265903.
Cf. A265332 (corresponding other index).
Cf. A000079 (positions of ones).
Cf. A000225 (from the term 3 onward the positions of 2's).
Cf. A000325 (from its third term 5 onward the positions of 3's, which occur always as the last term before the next descending subsequence of powers of two).

Programs

  • Mathematica
    terms = 100;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
t = Array[h, 2*terms];
A051135 = Take[Transpose[Tally[t]][[2]], terms];
b[_] = 1;
a[n_] := a[n] = With[{t = If[n == 1, 1, A051135[[n]]]}, b[t]++];
Array[a, terms] (* Jean-François Alcover, Dec 19 2021, after Robert G. Wilson v in A051135 *)

Formula

Let b(1) = 1, b(n) = A051135(n) for n > 1. Then a(n) is the number of b(k) that equal b(n) for 1 <= k <= n: sum( 1, 1<=k<=n and a(k)=a(n) ).
If A265332(n) = 1, then a(n) = A004001(n), otherwise a(n) = a(A080677(n)-1) = a(n - A004001(n)). - Antti Karttunen, Jan 09 2016

Extensions

Name amended by Antti Karttunen, Jan 09 2016

A276441 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = 1 + 2*a(n), a(A088359(n)) = 2*a(n), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 8, 13, 10, 9, 11, 31, 30, 28, 24, 16, 29, 26, 20, 25, 18, 17, 27, 22, 21, 19, 23, 63, 62, 60, 56, 48, 32, 61, 58, 52, 40, 57, 50, 36, 49, 34, 33, 59, 54, 44, 53, 42, 41, 51, 38, 37, 35, 55, 46, 45, 43, 39, 47, 127, 126, 124, 120, 112, 96, 64, 125, 122, 116, 104, 80, 121, 114, 100, 72, 113, 98
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276442.
Cf. A000079, A000225, A004001, A080677, A087686, A088359, A093879.
Related or similar permutations: A006068, A054429, A233275, A233277, A267111, A276343, A276345, A276443.
Cf. also arrays A265901, A265903.

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 1 + 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 2*a(A004001(n)-1).
As a composition of other permutations:
a(n) = A054429(A267111(n)).
a(n) = A233277(A276343(n)).
a(n) = A233275(A276345(n)).
a(n) = A006068(A276443(n)).
Other identities. For all n >= 1:
a(A000079(n-1)) = A000225(n).

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

Views

Author

Antti Karttunen, Dec 29 2015

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A267103 Row 3 of A265903; numbers that occur exactly three times in A004001.

Original entry on oeis.org

4, 15, 27, 30, 48, 54, 57, 61, 86, 96, 102, 105, 112, 115, 119, 124, 157, 172, 182, 188, 191, 202, 208, 211, 218, 221, 225, 233, 236, 240, 245, 251, 293, 314, 329, 339, 345, 348, 364, 374, 380, 383, 394, 400, 403, 410, 413, 417, 429, 435, 438, 445, 448, 452, 460, 463, 467, 472, 481, 484, 488, 493, 499, 506, 558
Offset: 1

Views

Author

Antti Karttunen, Jan 18 2016

Keywords

Comments

Numbers n for which A051135(n) = 3.

Links

Crossrefs

Column 3 of A265901, row 3 of A265903.
Cf. A004001, A051135, A087686, A188163, A266109.

Formula

a(n) = A087686(1+A266109(n)) = A087686(1+A087686(1+A188163(n))).
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