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

1,2

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A283167 A355807 A355808 * A341707 A347820 A318569

Adjacent sequences: A080076 A080077 A080078 * A080080 A080081 A080082

Keyword

nonn,base

Author

Reinhard Zumkeller, Jan 26 2003

Status

approved