A266188 -id:A266188 - cp's OEIS frontend

A087686 Elements of A004001 that repeat consecutively.

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

Roger L. Bagula, Sep 27 2003

nonn

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)

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

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.

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

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]

Other identities. For all n >= 1:

A080677(a(n)) = n. [See comments in A080677.]

A080677 a(n) = n + 1 - A004001(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 03 2003

nonn

From Antti Karttunen, Jan 10 2016: (Start)
This is the sequence b(n) mentioned on page 229 (page 5 of PDF) in Kubo & Vakil paper, but using starting offset 1 instead of 2.
The recursive sum formula for A004001, a(n) = a(a(n-1)) + a(n-a(n-1)) can be written also as a(n) = a(a(n-1)) + a(A080677(n-1)).
This is the least monotonic left inverse for sequence A087686. Proof: Taking the first differences of this sequence yields the characteristic function for the complement of A188163, because A188163 gives the positions where A004001 increases, and this sequence increases by one whenever A004001 does not increase (and vice versa). Sequence A188163 is also 1 followed by A088359 (see comment in former), whose complement A087686 is, thus A087686 is also the complement of A188163, apart from the initial one. Note also how A087686 is closed with respect to A004001 (see A266188).
(End)

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.

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences

Cf. A004001, A087686, A088359, A162598, A188163, A265332, A266188, A267111.

(define (A080677 n) (- (+ 1 n) (A004001 n))) ;; Scheme code for A004001 given in that entry. - Antti Karttunen, Jan 10 2016

a(n) = n + 1 - A004001(n).
Other identities. For all n >= 1:
a(A087686(n)) = n. [See comments.] - Antti Karttunen, Jan 10 2016

A266399 a(n) = A188163(A088359(n)); positions where A004001 obtains unique values.

Original entry on oeis.org

5, 9, 10, 17, 18, 19, 22, 33, 34, 35, 36, 39, 40, 43, 49, 65, 66, 67, 68, 69, 72, 73, 74, 77, 78, 81, 87, 88, 91, 97, 107, 129, 130, 131, 132, 133, 134, 137, 138, 139, 140, 143, 144, 145, 148, 149, 152, 158, 159, 160, 163, 164, 167, 173, 174, 177, 183, 193, 194, 197, 203, 213, 228, 257, 258, 259, 260, 261

Offset: 1

Antti Karttunen, Jan 18 2016

nonn

Numbers n for which A004001(n-1) < A004001(n) < A004001(n+1).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A004001.
Subsequence of A088359 and A188163.
Cf. also A266188.

a(n) = A188163(A088359(n)) = A088359(A088359(n)-1) = A188163(A188163(1+n)).
Other identities. For all n >= 1:
A004001(a(n)) = A088359(n).

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A087686 Elements of A004001 that repeat consecutively.

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A080677 a(n) = n + 1 - A004001(n).

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A266399 a(n) = A188163(A088359(n)); positions where A004001 obtains unique values.

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