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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A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

On Jul 15 1988 during a colloquium talk at Bell Labs, John Conway stated that he could prove that a(n)/n -> 1/2 as n approached infinity, but that the proof was extremely difficult. He therefore offered $100 to someone who could find an n_0 such that for all n >= n_0, we have |a(n)/n - 1/2| < 0.05, and he offered $10,000 for the least such n_0. I took notes (a scan of my notebook pages appears below), plus the talk - like all Bell Labs Colloquia at that time - was recorded on video. John said afterwards that he meant to say $1000, but in fact he said $10,000. I was in the front row. The prize was claimed by Colin Mallows, who agreed not to cash the check. - N. J. A. Sloane, Oct 21 2015
a(n) - a(n-1) = 0 or 1 (see the D. Newman reference). - Emeric Deutsch, Jun 06 2005
a(A188163(n)) = n and a(m) < n for m < A188163(n). - Reinhard Zumkeller, Jun 03 2011
From Daniel Forgues, Oct 04 2019: (Start)
Conjectures:
a(n) = n/2 iff n = 2^k, k >= 1.
a(n) = 2^(k-1): k times, for n = 2^k - (k-1) to 2^k, k >= 1. (End)

Examples

			If n=4, 2^4=16, a(16-i) = 2^(4-1) = 8 for 0 <= i <= 4-1 = 3, hence a(16)=a(15)=a(14)=a(13)=8.

References

  • J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
  • B. W. Conolly, Meta-Fibonacci sequences, in S. Vajda, editor, "Fibonacci and Lucas Numbers and the Golden Section", Halstead Press, NY, 1989, pp. 127-138.
  • R. K. Guy, Unsolved Problems Number Theory, Sect. E31.
  • D. R. Hofstadter, personal communication.
  • C. A. Pickover, Wonders of Numbers, "Cards, Frogs and Fractal sequences", Chapter 96, pp. 217-221, Oxford Univ. Press, NY, 2000.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Links

Crossrefs

Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences), A265332, A266341, A055748 (a chaotic cousin), A188163 (greedy inverse).
Cf. A004074 (A249071), A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Cf. also permutations A267111-A267112 and arrays A265901, A265903.

Programs

  • Haskell
    a004001 n = a004001_list !! (n-1)
a004001_list = 1 : 1 : h 3 1  {- memoization -}
  where h n x = x' : h (n + 1) x'
          where x' = a004001 x + a004001 (n - x)
-- Reinhard Zumkeller, Jun 03 2011
  • Magma
    [n le 2 select 1 else Self(Self(n-1))+ Self(n-Self(n-1)):n in [1..75]]; // Marius A. Burtea, Aug 16 2019
  • Maple
    A004001 := proc(n) option remember; if n<=2 then 1 else procname(procname(n-1)) +procname(n-procname(n-1)); fi; end;
  • Mathematica
    a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[ a[n], {n, 1, 75}] (* Robert G. Wilson v *)
  • PARI
    a=vector(100);a[1]=a[2]=1;for(n=3,#a,a[n]=a[a[n-1]]+a[n-a[n-1]]);a \\ Charles R Greathouse IV, Jun 10 2011
  • PARI
    first(n)=my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); v \\ Charles R Greathouse IV, Feb 26 2017
  • Python
    def a004001(n):
    A = {1: 1, 2: 1}
    c = 1 #counter
    while n not in A.keys():
        if c not in A.keys():
            A[c] = A[A[c-1]] + A[c-A[c-1]]
        c += 1
    return A[n]
# Edward Minnix III, Nov 02 2015
  • SageMath
    @CachedFunction
def a(n): # a = A004001
    if n<3: return 1
    else: return a(a(n-1)) + a(n-a(n-1))
[a(n) for n in range(1,101)] # G. C. Greubel, Apr 25 2024
  • Scheme
    ;; An implementation of memoization-macro definec can be found for example from: http://oeis.org/wiki/Memoization
(definec (A004001 n) (if (<= n 2) 1 (+ (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; Antti Karttunen, Oct 22 2014

Formula

Limit_{n->infinity} a(n)/n = 1/2 and as special cases, if n > 0, a(2^n-i) = 2^(n-1) for 0 <= i < = n-1; a(2^n+1) = 2^(n-1) + 1. - Benoit Cloitre, Aug 04 2002 [Corrected by Altug Alkan, Apr 03 2017]

A087686 Elements of A004001 that repeat consecutively.

Original entry on oeis.org

1, 2, 4, 7, 8, 12, 14, 15, 16, 21, 24, 26, 27, 29, 30, 31, 32, 38, 42, 45, 47, 48, 51, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 71, 76, 80, 83, 85, 86, 90, 93, 95, 96, 99, 101, 102, 104, 105, 106, 109, 111, 112, 114, 115, 116, 118, 119, 120, 121, 123, 124, 125, 126, 127
Offset: 1

Views

Author

Roger L. Bagula, Sep 27 2003

Keywords

Comments

Complement of A088359; A051135(a(n)) > 1. [Reinhard Zumkeller, Jun 03 2011]
From Antti Karttunen, Jan 18 2016: (Start)
This set of numbers is closed with respect to A004001, see A266188.
After 1, one more than the positions of zeros in A093879.
(End)

Links

Crossrefs

Cf. A004001, A051135, A093879.
Cf. A088359 (complement), A188163 (almost complement).
Cf. A080677 (the least monotonic left inverse).
Cf. also A266188, A265901, A265903, A267110, A267111, A267112.

Programs

  • Haskell
    import Data.List (findIndices)
a087686 n = a087686_list !! (n-1)
a087686_list = map succ $ findIndices (> 1) a051135_list
-- Reinhard Zumkeller, Jun 03 2011
(Scheme, with Antti Karttunen's IntSeq-library)
(define A087686 (MATCHING-POS 1 1 (lambda (n) (> (A051135 n) 1))))
;; Antti Karttunen, Jan 18 2016
  • Mathematica
    Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 digits=1000 a=Table[Conway[n], {n, 1, digits}]; b=Table[If[a[[n]]-a[[n-1]]==0, a[[n]], 0], {n, 2, digits}]; c=Delete[Union[b], 1]

Formula

Other identities. For all n >= 1:
A080677(a(n)) = n. [See comments in A080677.]

A088359 Numbers which occur only once in A004001.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 46, 49, 50, 52, 55, 59, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 87, 88, 89, 91, 92, 94, 97, 98, 100, 103, 107, 108, 110, 113, 117, 122, 129, 130, 131, 132
Offset: 1

Views

Author

Robert G. Wilson v, Sep 26 2003

Keywords

Comments

Out of the first one million terms (a(10^6) = 510403), 258661 occur only once.
Complement of A087686; A051135(a(n)) = 1. - Reinhard Zumkeller, Jun 03 2011
From Antti Karttunen, Jan 18 2016: (Start)
In general, out of the first 2^(n+1) terms of A004001, 2^(n-1) - 1 terms (a quarter) occur only once. See also illustration in A265332.
One more than the positions of ones in A093879.
(End)

Links

Crossrefs

Positions of ones in A051135.
Cf. A188163 (same sequence with prepended 1).
Cf. A087686 (complement).
Cf. A004001, A093879, A265332, A266399.
Cf. also A267110, A267111, A267112.

Programs

  • Haskell
    import Data.List (elemIndices)
a088359 n = a088359_list !! (n-1)
a088359_list = map succ $ elemIndices 1 a051135_list
-- Reinhard Zumkeller, Jun 03 2011
(Scheme, with Antti Karttunen's IntSeq-library)
(define A088359 (ZERO-POS 1 1 (COMPOSE -1+ A051135)))
;; Antti Karttunen, Jan 18 2016
  • Mathematica
    a[1] = 1; a[2] = 1; a[n_] := a[n] = a[ a[n - 1]] + a[n - a[n - 1]]; hc = Table[ a[n], {n, 1, 261}]; RunLengthEncodeOne[x_List] := Length[ # ] == 1 & /@ Split[x]; r = RunLengthEncodeOne[hc]; Select[ Range[ Length[r]], r[[ # ]] == True &]

Formula

From Antti Karttunen, Jan 18 2016: (Start)
Other identities.
For all n >= 0, a(A000079(n)) = A000051(n+1), that is, a(2^n) = 2^(n+1) + 1.
For all n >= 1:
a(n) = A004001(A266399(n)).
(End)

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2*a(n-1), a(A088359(n)) = 1+2*a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 15, 10, 13, 14, 12, 16, 17, 19, 23, 31, 18, 21, 27, 22, 29, 30, 20, 25, 26, 28, 24, 32, 33, 35, 39, 47, 63, 34, 37, 43, 55, 38, 45, 59, 46, 61, 62, 36, 41, 51, 42, 53, 54, 44, 57, 58, 60, 40, 49, 50, 52, 56, 48, 64, 65, 67, 71, 79, 95, 127, 66, 69, 75, 87, 111, 70, 77, 91, 119, 78, 93, 123
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2016

Keywords

Links

Crossrefs

Inverse: A267112.
Cf. A004001, A080677, A087686, A088359, A093879, A267108, A267109.
Similar or related permutations: A006068, A054429, A276441, A233275, A233277, A276343, A276345, A276445.
Cf. also permutations A266411, A266412 and arrays A265901, A265903.

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 1 + 2*a(A004001(n)-1).
Equally, for n > 1, if A093879(n-1) = 0, a(n) = 2*a(n - A004001(n)), otherwise a(n) = 1 + 2*a(A004001(n)-1). [Above formula in a more symmetric form.]
As a composition of other permutations:
a(n) = A054429(A276441(n)).
a(n) = A233275(A276343(n)).
a(n) = A233277(A276345(n)).
a(n) = A006068(A276445(n)).
Other identities. For all n >= 0:
a(2^n) = 2^n. [Follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper.]

A051135 a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001.

Original entry on oeis.org

2, 2, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3
Offset: 1

Views

Author

Robert Lozyniak (11(AT)onna.com)

Keywords

Comments

If the initial 2 is changed to a 1, the resulting sequence (A265332) has the property that if all 1's are deleted, the remaining terms are the sequence incremented. - Franklin T. Adams-Watters, Oct 05 2006
a(A088359(n)) = 1 and a(A087686(n)) > 1; first differences of A188163. - Reinhard Zumkeller, Jun 03 2011
From Robert G. Wilson v, Jun 07 2011: (Start)
a(k)=1 for k = 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, ..., ; (A088359)
a(k)=2 for k = 1, 2, 7, 12, 14, 21, 24, 26, 29, 38, 42, 45, 47, 51, 53, ..., ; (1 followed by A266109)
a(k)=3 for k = 4, 15, 27, 30, 48, 54, 57, 61, 86, 96, 102, 105, 112, ..., ; (A267103)
a(k)=4 for k = 8, 31, 58, 62, 106, 116, 120, 125, 192, 212, 222, 226, ..., ;
a(k)=5 for k = 16, 63, 121, 126, 227, 242, 247, 253, 419, 454, 469, ..., ;
a(k)=6 for k = 32, 127, 248, 254, 475, 496, 502, 509, 894, 950, 971, ..., ;
a(k)=7 for k = 64, 255, 503, 510, 978, 1006, 1013, 1021, 1872, 1956, ..., ;
a(k)=8 for k = 128, 511, 1014, 1022, 1992, 2028, 2036, 2045, 3864, ..., ;
a(k)=9 for k = 256, 1023, 2037, 2046, 4029, 4074, 4083, 4093, 7893, ..., ;
a(k)=10 for k = 512, 2047, 4084, 4094, 8113, 8168, 8178, 8189, ..., . (End)
Compare above to array A265903. - Antti Karttunen, Jan 18 2016

Links

Crossrefs

Cf. A004001, A087686, A093879, A188163, A266109, A267103 and array A265903.
Cf. A088359 (positions of ones).
Cf. A265332 (essentially the same sequence, but with a(1) = 1 instead of 2).

Programs

  • Haskell
    import Data.List (group)
a051135 n = a051135_list !! (n-1)
a051135_list = map length $ group a004001_list
-- Reinhard Zumkeller, Jun 03 2011
  • Magma
    nmax:=200;
h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..5*nmax]]; // h = A004001
A188163:= function(n)
   for j in [1..3*nmax+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
A051135:= func< n | A188163(n+1) - A188163(n) >;
[A051135(n): n in [1..nmax]]; // G. C. Greubel, May 20 2024
  • Maple
    a[1]:=1: a[2]:=1: for n from 3 to 300 do a[n]:=a[a[n-1]]+a[n-a[n-1]] od: A:=[seq(a[n],n=1..300)]:for j from 1 to A[nops(A)-1] do c[j]:=0: for n from 1 to 300 do if A[n]=j then c[j]:=c[j]+1 else fi od: od: seq(c[j],j=1..A[nops(A)-1]); # Emeric Deutsch, Jun 06 2006
  • Mathematica
    a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Array[a, 250]; Take[ Transpose[ Tally[t]][[2]], 105] (* Robert G. Wilson v, Jun 07 2011 *)
  • SageMath
    @CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1)) # h=A004001
def A188163(n):
    for j in range(1,2*n+1):
        if h(j)==n: return j
def A051135(n): return A188163(n+1) - A188163(n)
[A051135(n) for n in range(1,201)] # G. C. Greubel, May 20 2024
  • Scheme
    (define (A051135 n) (- (A188163 (+ 1 n)) (A188163 n))) ;; Antti Karttunen, Jan 18 2016

Formula

From Antti Karttunen, Jan 18 2016: (Start)
a(n) = A188163(n+1) - A188163(n). [after Reinhard Zumkeller's Jun 03 2011 comment above]
Other identities:
a(n) = 1 if and only if A093879(n-1) = 1. [See A188163 for a reason.]
(End)

Extensions

More terms from Jud McCranie
Added links (in parentheses) to recently submitted related sequences - Antti Karttunen, Jan 18 2016

A188163 Smallest m such that A004001(m) = n.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 46, 49, 50, 52, 55, 59, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 87, 88, 89, 91, 92, 94, 97, 98, 100, 103, 107, 108, 110, 113, 117, 122
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 03 2011

Keywords

Comments

How is this related to A088359? - R. J. Mathar, Jan 09 2013
It is not hard to show that a(n) exists for all n, and in particular a(n) < 2^n. - Charles R Greathouse IV, Jan 13 2013
From Antti Karttunen, Jan 10 & 18 2016: (Start)
Positions of records in A004001. After 1 the positions where A004001 increases (by necessity by one).
An answer to the question of R. J. Mathar above: This sequence is equal to A088359 with prepended 1. This follows because at each of its unique values (terms of A088359), A004001 must grow, but it can grow nowhere else. See Kubo and Vakil paper and especially the illustrations of Q and R-trees on pages 229-230 (pages 5 & 6 in PDF) and also in sequence A265332.
Obviously A004001 can obtain unique values only at points which form a subset (A266399) of this sequence.
(End)

Links

Crossrefs

Equal to A088359 with prepended 1.
Column 1 of A265901, Row 1 of A265903.
Cf. A051135 (first differences).
Cf. A087686 (complement, apart from the initial 1).
Cf. A004001 (also the least monotonic left inverse of this sequence).
Cf. A093879, A265332.
Cf. A266399 (a subsequence).

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a188163 n = succ $ fromJust $ elemIndex n a004001_list
  • Magma
    h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..500]]; // h=A004001
A188163:= function(n)
   for j in [1..2*n+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
[A188163(n): n in [1..100]]; // G. C. Greubel, May 20 2024
  • Maple
    A188163 := proc(n)
    for a from 1 do
        if A004001(a) = n then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, May 15 2013
  • Mathematica
    h[1] = 1; h[2] = 1; h[n_] := h[n] = h[h[n-1]] + h[n - h[n-1]];
a[n_] := For[m = 1, True, m++, If[h[m] == n, Return[m]]];
Array[a, 64] (* Jean-François Alcover, Jan 27 2018 *)
  • SageMath
    @CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1)) # h=A004001
def A188163(n):
    for j in range(1,2*n+2):
        if h(j)==n: return j
[A188163(n) for n in range(1,101)] # G. C. Greubel, May 20 2024
  • Scheme
    (define A188163 (RECORD-POS 1 1 A004001))
;; Code for A004001 given in that entry. Uses also my IntSeq-library. - Antti Karttunen, Jan 18 2016

Formula

Other identities. For all n >= 1:
A004001(a(n)) = n and A004001(m) < n for m < a(n).
A051135(n) = a(n+1) - a(n).

A214546 First differences of A140472.

Original entry on oeis.org

1, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 8, -7, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 16, -15, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 8, -7, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 32, -31, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 20 2012

Keywords

Comments

a(n) = A140472(n+1) - A140472(n);
a(A016825(n)) = 0; a(A042965(n)) <> 0;
for n > 0: a(A008586(n)) < 0, a(A005843(n)) <= 0, a(A042968(n)) >= 0;
a(A004273(n)) > 0.

Links

Crossrefs

Cf. A093879.

Programs

  • Haskell
    a214546 n = a214546_list !! n
a214546_list = zipWith (-) (tail a140472_list) a140472_list

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).

A276445 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A001969(1+a(n-1)), a(A088359(n)) = A000069(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 12, 13, 14, 8, 15, 11, 9, 10, 24, 25, 26, 28, 16, 27, 31, 22, 29, 19, 17, 30, 21, 23, 18, 20, 48, 49, 50, 52, 56, 32, 51, 55, 62, 44, 53, 59, 38, 57, 35, 33, 54, 61, 42, 63, 47, 45, 58, 37, 39, 34, 60, 41, 43, 46, 36, 40, 96, 97, 98, 100, 104, 112, 64, 99, 103, 110, 124, 88, 101, 107, 118, 76, 105, 115, 70
Offset: 1

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Author

Antti Karttunen, Sep 03 2016

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Crossrefs

Inverse: A276446.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A267111, A276443 (compare the scatter plots).

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A001969(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A000069(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A267111(n)).

A276343 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A005187(1+a(n)), a(A088359(n)) = A055938(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, 5, 4, 15, 14, 13, 12, 11, 9, 10, 8, 31, 30, 29, 28, 27, 26, 24, 20, 25, 21, 23, 22, 17, 18, 19, 16, 63, 62, 61, 60, 59, 58, 57, 55, 51, 43, 56, 52, 44, 54, 48, 53, 50, 45, 36, 47, 37, 39, 49, 40, 41, 46, 42, 33, 34, 35, 38, 32, 127, 126, 125, 124, 123, 122, 121, 120, 118, 114, 106, 90, 119, 115, 107, 91, 117, 111, 99
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276344.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276345, A276441.
Compare also to the scatter-plots of A276443 and A276445.

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A005187(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A055938(a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A233276(A267111(n)).
a(n) = A233278(A276441(n)).
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