A088359 Numbers which occur only once in A004001.

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

Author

Robert G. Wilson v, Sep 26 2003

Keywords

• nonn

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

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

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)