A285097 a(n) = difference between the positions of two least significant 1-bits in base-2 representation of n, or 0 if there are less than two 1-bits in n (when n is either zero or a power of 2).

0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 1, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 1, 4, 2, 1, 1, 3, 3, 2, 1, 1, 2, 1, 1, 2, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 4, 1, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1

Offset

0,6

Comments

a(1+n) is the length of the least significant run of 0-bits in n, or 0 if n is one of terms of A000225. - Antti Karttunen, Oct 14 2023

Links

Antti Karttunen, Table of n, a(n) for n = 0..8192

Index entries for sequences related to binary expansion of n

Formula

If A000120(n) < 2, a(n) = 0, otherwise a(n) = A285099(n) - A007814(n) = A007814(A129760(n)) - A007814(n).

a(n) = 0 if n is 0 or of the form 2^k, (k>=0), otherwise a(n) = v_2(A000265(n)-1), where v_2(i) = A007814(i). - Ridouane Oudra, Oct 20 2019

Example

For n = 3, "11" in binary, the second least significant 1-bit (the second 1-bit from the right) is at position 1 and the rightmost 1-bit is at position 0), thus a(3) = 1-0 = 1.
For n = 4, "100" in binary, there is just one 1-bit present, thus a(4) = 0.
For n = 5, "101" in binary, the second 1-bit from the right is at position 2, and the least significant 1 is at position 0, thus a(5) = 2-0 = 2.
For n = 26, "11010" in binary, the second 1-bit from the right is at position 3, and the least significant 1 is at position 1, thus a(26) = 3-1 = 2.

Mathematica

a007814[n_]:=IntegerExponent[n, 2]; a285099[n_]:=If[DigitCount[n, 2, 1]<2, 0, a007814[BitAnd[n, n - 1]]]; a[n_]:=If[DigitCount[n, 2, 1]<2, 0, a285099[n] - a007814[n]]; Table[a[n], {n, 0, 150}] (* Indranil Ghosh, Apr 20 2017 *)

Prog

(Scheme) (define (A285097 n) (if (<= (A000120 n) 1) 0 (- (A285099 n) (A007814 n))))
(Python)
import math
def a007814(n): return int(math.log(n - (n & n - 1), 2))
def a285099(n): return 0 if bin(n)[2:].count("1") < 2 else a007814(n & (n - 1))
def a(n): return 0 if bin(n)[2:].count("1")<2 else a285099(n) - a007814(n) # Indranil Ghosh, Apr 20 2017
(PARI) A285097(n) = if(!n || !bitand(n, n-1), 0, valuation((n>>valuation(n, 2))-1, 2)); \\ Antti Karttunen, Oct 14 2023

Crossrefs

Cf. A000120, A000225, A000265, A007814, A119387, A129760, A285099, A366370.

Sequence in context: A171846 A097230 A144789 * A279209 A087117 A029340

Adjacent sequences: A285094 A285095 A285096 * A285098 A285099 A285100

Keyword

nonn,base

Author

Antti Karttunen, Apr 20 2017

Status

approved