A087686 -id:A087686 - cp's OEIS frontend

A276345 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A055938(a(n)), a(A088359(n)) = A005187(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

1, 2, 3, 5, 4, 7, 6, 12, 10, 8, 15, 9, 11, 14, 13, 27, 23, 19, 16, 31, 21, 18, 22, 17, 26, 30, 20, 25, 24, 29, 28, 58, 53, 46, 38, 32, 63, 48, 41, 35, 42, 40, 34, 50, 33, 57, 62, 44, 39, 49, 37, 47, 45, 36, 56, 55, 61, 43, 54, 52, 51, 60, 59, 121, 113, 104, 89, 74, 64, 127, 108, 95, 81, 70, 82, 93, 79, 67, 98, 77, 66, 112, 65, 120

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276346.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276343, A276441.

(definec (A276345 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A055938 (A276345 (+ -1 (A080677 n))))) (else (A005187 (+ 1 (A276345 (+ -1 (A004001 n))))))))

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

A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 12, 9, 13, 8, 15, 14, 11, 19, 24, 22, 18, 27, 21, 23, 17, 29, 28, 25, 16, 31, 30, 26, 20, 36, 45, 43, 40, 54, 51, 35, 49, 42, 39, 41, 58, 48, 53, 34, 52, 38, 50, 44, 61, 60, 33, 59, 56, 55, 46, 32, 63, 62, 57, 47, 37, 69, 83, 81, 78, 102, 99, 74, 97, 93, 91, 68, 116, 112, 80, 88, 77, 109, 73, 75, 96, 90

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276345.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276344, A276442.

(definec (A276346 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A088359 (A276346 (- (A213714 n) 1)))) (else (A087686 (+ 1 (A276346 (A234017 n)))))))

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A088359(a(A213714(n)-1)), otherwise a(n) = A087686(1+a(A234017(n))).
As a composition of other permutations:
a(n) = A276442(A233275(n)).
a(n) = A267112(A233277(n)).

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 15, 13, 14, 16, 17, 18, 20, 24, 19, 23, 30, 21, 27, 25, 22, 29, 31, 26, 28, 32, 33, 34, 36, 40, 48, 35, 39, 46, 60, 37, 43, 54, 41, 51, 49, 38, 45, 58, 47, 63, 61, 42, 53, 55, 50, 44, 57, 59, 62, 52, 56, 64, 65, 66, 68, 72, 80, 96, 67, 71, 78, 92, 120, 69, 75, 86, 108, 73, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276444.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A276441, A276445 (compare the scatter plots).

(definec (A276443 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A000069 (+ 1 (A276443 (+ -1 (A080677 n)))))) (else (A001969 (+ 1 (A276443 (+ -1 (A004001 n))))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A000069(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A001969(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A276441(n)).

A266188 a(n) = A004001(A087686(n)).

Original entry on oeis.org

1, 1, 2, 4, 4, 7, 8, 8, 8, 12, 14, 15, 15, 16, 16, 16, 16, 21, 24, 26, 27, 27, 29, 30, 30, 31, 31, 31, 32, 32, 32, 32, 32, 38, 42, 45, 47, 48, 48, 51, 53, 54, 54, 56, 57, 57, 58, 58, 58, 60, 61, 61, 62, 62, 62, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 71, 76, 80, 83, 85, 86, 86, 90, 93, 95, 96, 96, 99, 101, 102, 102, 104

Offset: 1

Antti Karttunen, Jan 10 2016

nonn

Discarding duplicates gives A087686 back, i.e., this set of numbers is closed with respect to A004001.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A004001, A080677, A087686.

(define (A266188 n) (A004001 (A087686 n)))

a(n) = A004001(A087686(n)).

A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 14, 15, 13, 16, 17, 18, 21, 19, 24, 27, 22, 20, 26, 30, 25, 31, 28, 23, 29, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 64, 65, 66, 71, 67, 76, 86, 72, 68, 80, 96, 77, 106, 87, 73, 90, 69, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276443.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A276442, A276446.

(definec (A276444 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A088359 (A276444 (A245710 n)))) (else (A087686 (+ 1 (A276444 (- (A115384 n) 1)))))))

a(1) = 1, and for n > 1, if A010060(n) = 0 [when n is one of the evil numbers, A001969], a(n) = A088359(a(A245710(n))), otherwise a(n) = A087686(1+a(A115384(n)-1)).
As a composition of other permutations:
a(n) = A276442(A006068(n)).

A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(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, 11, 14, 15, 13, 8, 9, 10, 12, 20, 26, 30, 25, 31, 28, 23, 29, 16, 17, 18, 21, 19, 24, 27, 22, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 70, 85, 105, 84, 120, 103, 82, 104, 126, 117, 98, 118, 79, 101, 119, 100, 127, 122, 108, 123, 89

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276445.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A267112, A276444.

(definec (A276446 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A087686 (+ 1 (A276446 (A245710 n))))) (else (A088359 (A276446 (- (A115384 n) 1))))))

a(1) = 1, and for n > 1, if A010060(n) = 1 [when n is one of the odious numbers, A000069], a(n) = A088359(a(A115384(n)-1)), otherwise a(n) = A087686(1+a(A245710(n))).
As a composition of other permutations:
a(n) = A267112(A006068(n)).

A269851 a(0) = 1, a(A087686(1+n)) = 2*a(n), a(A088359(n)) = A250469(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 5, 6, 16, 21, 25, 7, 18, 15, 10, 12, 32, 45, 55, 49, 11, 42, 51, 35, 50, 27, 14, 36, 33, 30, 20, 24, 64, 93, 115, 91, 121, 13, 90, 123, 125, 77, 110, 147, 65, 98, 39, 22, 84, 105, 85, 102, 87, 70, 100, 57, 54, 28, 72, 69, 66, 60, 40, 48, 128, 189, 235, 203, 187, 169, 17, 186, 267, 305, 217, 143, 230

Offset: 0

Antti Karttunen, Mar 07 2016

nonn

Permutation of natural numbers obtained from the sieve of Eratosthenes, combined with the permutation obtained from Hofstadter-Conway $10000 sequence (A004001). Note the indexing: Domain starts from 0, range from 1.

Inverse: A269852.
Cf. A004001, A087686, A088359, A093879, A250469.
Related or similar permutations: A252755, A267111, A269855.

a(0) = 1, a(1) = 2, for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(n - A004001(n)), otherwise [when n is in A088359], a(n) = A250469(a(A004001(n)-1)).
As a composition of related permutations:
a(n) = A252755(A267111(n)).
Other identities. For all n >= 0:
a(2^n) = 2^(n+1).

A269852 Permutation of natural numbers: a(1) = 0, after which, a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(A268674(2n+1))).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 11, 4, 5, 14, 20, 15, 37, 26, 13, 8, 70, 12, 135, 30, 9, 47, 264, 31, 10, 85, 25, 57, 521, 29, 1034, 16, 28, 156, 23, 27, 2059, 292, 46, 62, 4108, 21, 8205, 105, 17, 557, 16398, 63, 19, 24, 22, 191, 32783, 56, 18, 120, 55, 1079, 65552, 61, 131089, 2114, 84, 32, 44, 60, 262162, 348, 59, 53

Offset: 1

Antti Karttunen, Mar 07 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

Inverse: A269851.
Cf. A004001, A087686, A088359, A268674.
Related or similar permutations: A252756, A267112, A269856.

a(1) = 0; after which, for even n, a(n) = A087686(1+a(n/2)), and for odd n, a(n) = A088359(a(A268674(n))).
Other identities. For all n >= 1:
a(2^n) = 2^(n-1).
As a composition of other permutations:
a(n) = A267112(A252756(n)).

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

N. J. A. Sloane

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)

			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.

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.

T. D. Noe, Table of n, a(n) for n = 1..10000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan and O. Ozgur Aybar, On Families of Solutions for Meta-Fibonacci Recursions Related to Hofstadter-Conway $10000 Sequence, Slides of talk at presented in 5th International Interdisciplinary Chaos Symposium on Chaos and Complex Systems, May 9-12, 2019.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, Volume 2017, Article ID 2614163, 8 pages.
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.
Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, Plot of 100000 terms of a(n)-n/2
D. R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, Oct 10 2014; Part 1, Part 2.
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
N. J. A. Sloane, Scan of notebook pages with notes on John Conway's Colloquium talk on Jul 15 1988 [See Comments above]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991
Eric Weisstein's World of Mathematics, Hofstadter-Conway 10000-Dollar Sequence.
Eric Weisstein's World of Mathematics, Newman-Conway Sequence
Wikipedia, Hofstadter sequence
Index entries for sequences related to binary expansion of n
Index entries for Hofstadter-type sequences

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.

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

[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

A004001 := proc(n) option remember; if n<=2 then 1 else procname(procname(n-1)) +procname(n-procname(n-1)); fi; end;

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 *)

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

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

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

@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

;; 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

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]

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

Robert G. Wilson v, Sep 26 2003

nonn

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)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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.

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

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 &]

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)

cp's OEIS Frontend

A276345 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A055938(a(n)), a(A088359(n)) = A005187(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A266188 a(n) = A004001(A087686(n)).

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A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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Formula

A269851 a(0) = 1, a(A087686(1+n)) = 2*a(n), a(A088359(n)) = A250469(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

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Formula

A269852 Permutation of natural numbers: a(1) = 0, after which, a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(A268674(2n+1))).

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Formula

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

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