A004001 -id:A004001 - cp's OEIS frontend

A283469 a(n) = A004001(A004001(n-1)) OR A004001(n-A004001(n-1)), a(1) = a(2) = 1.

1, 1, 1, 1, 3, 2, 2, 2, 3, 3, 7, 7, 4, 4, 4, 4, 5, 5, 7, 7, 7, 7, 14, 14, 15, 15, 15, 8, 8, 8, 8, 8, 9, 9, 11, 11, 13, 13, 14, 15, 14, 14, 15, 15, 15, 15, 15, 15, 15, 29, 29, 30, 30, 30, 31, 31, 31, 31, 16, 16, 16, 16, 16, 16, 17, 17, 19, 19, 21, 21, 21, 23, 22, 23, 26, 26, 27, 27, 27, 27, 30, 31, 31, 31, 31, 31, 31, 30, 31, 31, 31, 31, 31

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003986, A283470, A283472, A283473 (positions where coincides with A004001).

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[BitOr[a[#], a[n - #]] &@ a[n - 1], {n, 93}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283469 n) (if (<= n 2) 1 (A003986bi (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; A003986bi implements bitwise-OR (see A003986). Code for A004001 given under that entry.

a(1) = a(2) = 1; for n > 2, a(n) = A004001(A004001(n-1)) OR A004001(A080677(n-1)), where OR is bitwise-or (A003986)
Other identities. For all n >= 1:
a(n) = A283470(n) + A283472(n).
A004001(n) = a(n) + A283472(n).

A283471 Numbers n > 2 such that A004001(A004001(n-1)) = A004001(n-A004001(n-1)).

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 13, 14, 15, 16, 18, 28, 29, 30, 31, 32, 34, 59, 60, 61, 62, 63, 64, 66, 122, 123, 124, 125, 126, 127, 128, 130, 249, 250, 251, 252, 253, 254, 255, 256, 258, 504, 505, 506, 507, 508, 509, 510, 511, 512, 514, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1026, 2038, 2039, 2040, 2041, 2042, 2043, 2044, 2045

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Equally, numbers n > 2 for which A004001(A004001(n-1)) = A004001(A080677(n-1)).

Antti Karttunen, Table of n, a(n) for n = 1..185

Positions of zeros in A283468 and A283470.
Cf. A004001, A080677, A283473.
Subsequence of A283482.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Select[Range[3, 2^11], Function[n, a[#] == a[n - #] &@ a[n - 1]]] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

A283472 a(n) = A004001(A004001(n-1)) AND A004001(n-A004001(n-1)), a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 2, 2, 3, 0, 0, 4, 4, 4, 4, 4, 5, 4, 5, 5, 6, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 8, 8, 8, 10, 10, 10, 11, 11, 12, 12, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 16, 16, 17, 16, 17, 16, 17, 17, 16, 18, 18, 16, 16, 16, 17, 18, 18, 16, 16, 16, 17, 17, 17, 18, 20, 20, 20, 21, 22, 22, 24, 24, 24, 24

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004001, A004198, A283469, A283470, A283473 (positions of zeros).

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[BitAnd[a[#], a[n - #]] &@ a[n - 1] - Boole[n <= 2], {n, 97}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283472 n) (if (<= n 2) 0 (A004198bi (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; A004198bi implements bitwise-AND (see A004198). Code for A004001 given under that entry.

a(1) = a(2) = 0; for n > 2, a(n) = A004001(A004001(n-1)) AND A004001(A080677(n-1)), where AND is bitwise-and (A004198).
Other identities. For all n >= 1:
a(n) = A283469(n) - A283470(n).
A004001(n) = A283469(n) + a(n) = A283470(n) + 2*a(n).

A283473 Numbers n for which A004001(n) = A283470(n).

Original entry on oeis.org

1, 2, 5, 11, 12, 23, 24, 25, 26, 27, 50, 51, 52, 53, 54, 55, 56, 57, 58, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Equally, numbers n such that A004001(n) = A283469(n).

Antti Karttunen, Table of n, a(n) for n = 1..803

Positions of zeros in A283472.

Cf. A004001, A283469, A283470, A283471.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; With[{nn = 500}, Function[s, Select[Range@ nn, a@ # == s[[#]] &]]@ Table[BitXor[a[#], a[n - #]] &@ a[n - 1] + Boole[n <= 2], {n, nn}]] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

A317754 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A004001. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Altug Alkan, Aug 06 2018

Altug Alkan, Table of n, a(n) for n = 1..32768

Cf. A004001, A004074, A317648, A317742.

t:= proc(n) option remember; `if`(n<3, 1,
      t(t(n-1)) +t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<3, 1,
      n -b(t(n)) -b(n-t(n)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

Block[{t = NestWhile[Function[{a, n}, Append[a, a[[a[[-1]] ]] + a[[-a[[-1]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Last@ # < 60 &], b}, b = NestWhile[Function[{b, n}, Append[b, n - b[[t[[n]] ]] - b[[-t[[n]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Length@ # < Length@ t &]; Array[2 b[[#]] - # &, Length@ b] ] (* Michael De Vlieger, Aug 07 2018 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[n-t[n-1]]+t[t[n-1]]); b=vector(99); b[1]=b[2]=1; for(n=3, #b, b[n] = n-b[t[n]]-b[n-t[n]]); vector(99, k, 2*b[k]-k)

A249071 a(n) = A004001(2*n) - n, where A004001 is Hofstadter-Conway $10000 sequence.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 1, 0, 1, 2, 2, 3, 3, 4, 4, 3, 4, 4, 3, 3, 2, 2, 1, 0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 6, 6, 7, 7, 6, 7, 7, 6, 6, 5, 6, 6, 5, 5, 4, 4, 3, 3, 2, 1, 0, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 10, 11, 11, 12, 12, 11, 12, 12, 11, 11, 10, 11, 11, 12, 12, 11, 12, 12, 11, 11, 10, 11, 11, 10, 10, 9, 9, 8, 8, 8, 8, 8, 7, 7, 6, 5, 5, 4, 3, 3, 2, 1, 0

Offset: 1

Antti Karttunen, Oct 22 2014

nonn

Hofstadter shows the plot of function A004001(n)-(n/2) at time 10:52 during the part two of DIMACS lecture. This sequence is obtained as the bisection of that function, thus containing only integers. Cf. also A004074.

Antti Karttunen, Table of n, a(n) for n = 1..16384
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, October 10 2014; Part 1, Part 2.
Wikipedia, Blancmange curve
Index entries for Hofstadter-type sequences

Cf. A004001, A004074.

Cf. also A233270 (also has a similar Blancmange curve appearance).

a(n) = A004001(2*n) - n.

a(n) = A004074(2*n) / 2. [Also the even bisection of A004074 halved.]

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

Antti Karttunen, Sep 03 2016

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.

(definec (A276343 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A005187 (+ 1 (A276343 (+ -1 (A080677 n)))))) (else (A055938 (A276343 (+ -1 (A004001 n)))))))

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)).

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(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, 13, 14, 12, 11, 10, 9, 8, 31, 28, 29, 30, 23, 25, 27, 26, 22, 24, 21, 20, 19, 18, 17, 16, 63, 59, 60, 61, 50, 52, 62, 53, 55, 56, 58, 41, 44, 49, 57, 51, 46, 54, 48, 40, 43, 47, 45, 39, 42, 38, 37, 36, 35, 34, 33, 32, 127, 122, 123, 124, 108, 110, 125, 111, 113, 114, 126, 89, 92, 117, 115, 118, 94, 119, 121

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276343.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276346, A276442.

(definec (A276344 n) (cond ((< n 2) n) ((not (zero? (A079559 n))) (A087686 (+ 1 (A276344 (- (A213714 n) 1))))) (else (A088359 (A276344 (A234017 n))))))

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

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.

Original entry on oeis.org

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)).

cp's OEIS Frontend

A283469 a(n) = A004001(A004001(n-1)) OR A004001(n-A004001(n-1)), a(1) = a(2) = 1.

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A283471 Numbers n > 2 such that A004001(A004001(n-1)) = A004001(n-A004001(n-1)).

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A283472 a(n) = A004001(A004001(n-1)) AND A004001(n-A004001(n-1)), a(1) = a(2) = 0.

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A283473 Numbers n for which A004001(n) = A283470(n).

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Mathematica

A317754 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A004001. a(n) = 2*b(n) - n.

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PARI

A249071 a(n) = A004001(2*n) - n, where A004001 is Hofstadter-Conway $10000 sequence.

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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.

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A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

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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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