A003817 a(0) = 0, a(n) = a(n-1) OR n.

0, 1, 3, 3, 7, 7, 7, 7, 15, 15, 15, 15, 15, 15, 15, 15, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63

Offset

0,3

Comments

Also, 0+1+2+...+n in lunar arithmetic in base 2 written in base 10. - N. J. A. Sloane, Oct 02 2010

For n>0: replace all 0's with 1's in binary representation of n. - Reinhard Zumkeller, Jul 14 2003

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 [From Reinhard Zumkeller, Nov 14 2009]

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Ralf Stephan, Some divide-and-conquer sequences ...

Ralf Stephan, Table of generating functions

Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Reinhard Zumkeller, Logical Convolutions

Index entries for sequences related to dismal (or lunar) arithmetic

Formula

a(n) = a(n-1) + n*(1-floor(a(n-1)/n)). If 2^(k-1) <= n < 2^k, a(n) = 2^k - 1. - Benoit Cloitre, Aug 25 2002

a(n) = 1 + 2*a(floor(n/2)) for n > 0. - Benoit Cloitre, Apr 04 2003

G.f.: (1/(1-x)) * Sum_{k>=0} 2^k*x^2^k. - Ralf Stephan, Apr 18 2003

a(n) = 2*A053644(n)-1 = A092323(n) + A053644(n). - Reinhard Zumkeller, Feb 15 2004; corrected by Anthony Browne, Jun 26 2016

a(n) = OR{k OR (n-k): 0<=k<=n}. - Reinhard Zumkeller, Jul 15 2008

For n>0: a(n+1) = A035327(n) + n = A035327(n) XOR n. - Reinhard Zumkeller, Nov 14 2009

A092323(n+1) = floor(a(n)/2). - Reinhard Zumkeller, Jul 18 2010

a(n) = A265705(n,0) = A265705(n,n). - Reinhard Zumkeller, Dec 15 2015

a(n) = A062383(n) - 1.

G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x). - Ilya Gutkovskiy, Aug 31 2019

a(n) >= A175039(n) - Austin Shapiro, Dec 29 2022

Maple

A003817 := n -> n + Bits:-Nand(n, n):
seq(A003817(n), n=0..61); # Peter Luschny, Sep 23 2019

Mathematica

a[0] = 0; a[n_] := a[n] = BitOr[ a[n-1], n]; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Dec 19 2011 *)
nxt[{n_, a_}]:={n+1, BitOr[a, n+1]}; Transpose[NestList[nxt, {0, 0}, 70]] [[2]] (* Harvey P. Dale, May 06 2016 *)
2^BitLength[Range[0, 100]]-1 (* Paolo Xausa, Feb 08 2024 *)

Prog

(PARI) a(n)=1<<(log(2*n+1)\log(2))-1 \\ Charles R Greathouse IV, Dec 08 2011
(Haskell)
import Data.Bits ((.|.))
a003817 n = if n == 0 then 0 else 2 * a053644 n - 1
a003817_list = scanl (.|.) 0 [1..] :: [Integer]
-- Reinhard Zumkeller, Dec 08 2012, Jan 15 2012
(Python)
def a(n): return 0 if n==0 else 1 + 2*a(int(n/2)) # Indranil Ghosh, Apr 28 2017

Crossrefs

This is Guy Steele's sequence GS(6, 6) (see A135416).

Cf. A000004, A142149, A086099, A142150, A142151, A001477, A062383.

Cf. A167832, A167878. - Reinhard Zumkeller, Nov 14 2009

Cf. A179526; subsequence of A007448. - Reinhard Zumkeller, Jul 18 2010

Cf. A265705.

Sequence in context: A160515 A105670 A283996 * A092474 A225851 A107470

Adjacent sequences: A003814 A003815 A003816 * A003818 A003819 A003820

Keyword

nonn,base,nice

Author

Marc LeBrun

Status

approved