A080079 Least number causing the longest carry sequence when adding numbers <= n to n in binary representation.

1, 2, 1, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47

Offset: 1

Author

Reinhard Zumkeller, Jan 26 2003

Keywords

• nonn
• base

Comments

T(n,k) < T(n,a(n)) = A070940(n) for 1 <= k < a(n) and T(n,k) <= T(n,a(n)) for a(n) <= k <= n, where T is defined as in A080080.

a(n) gives the distance from n to the nearest 2^t > n. - Ctibor O. Zizka, Apr 09 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A327489, A004755, A062383, A080080, A240769, A006257.

Programs

Haskell

a080079 n = (length $ takeWhile (< a070940 n) (a080080_row n)) + 1
-- Reinhard Zumkeller, Apr 22 2013

Magma

[-n+2*2^Floor(Log(n)/Log(2)): n in [1..80]]; // Vincenzo Librandi, Dec 01 2016

Maple

# Alois P. Heinz observes in A327489:
A080079 := n -> 1 + Bits:-Nor(n, n):
# Likewise:
A080079 := n -> 1 + Bits:-Nand(n, n):
seq(A080079(n), n=1..81); # Peter Luschny, Sep 23 2019

Mathematica

Flatten@Table[Nest[Most[RotateRight[#]] &, Range[n], n - 1], {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[FromDigits[(IntegerDigits[n, 2] /. {0 -> 1, 1 -> 0}), 2] +
1, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1] + 1, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[2 2^Floor[Log[2, n]] - n, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Flatten@Table[Reverse@Range[2^n], {n, 0, 4}] (* Birkas Gyorgy, Feb 07 2011 *)

Python

def A080079(n): return (1 << n.bit_length())-n # Chai Wah Wu, Jun 30 2022

Formula

From Benoit Cloitre, Feb 22 2003: (Start)

a(n) = A004755(n) - 2*n.

a(n) = -n + 2*2^floor(log(n)/log(2)). (End)

From Ralf Stephan, Jun 02 2003: (Start)

a(n) = n iff n = 2^k, otherwise a(n) = A035327(n-1).

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

a(0) = 0, a(2*n) = 2*a(n), a(2*n+1) = 2*a(n)-1 + 2*[n==0]. - Ralf Stephan, Jan 04 2004

a(n) = A240769(n,1); A240769(n, a(n)) = 1. - Reinhard Zumkeller, Apr 13 2014

a(n) = n + 1 - A006257(n). - Reinhard Zumkeller, Apr 14 2014