A337319 a(n) = Sum_{i = 1..floor(log_2(n))+1} g(frac(n/2^i)), where g(t) = [0 if t = 0, -1 if 0 < t < 1/2, 1 if t >= 1/2], and where frac(x) denotes the fractional part.

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

Offset: 1

Author

Christoph B. Kassir, Aug 23 2020

Keywords

• sign

Comments

The function a(n) is a measure of how many times n rounds up (assigned value +1), down (assigned value -1), or not at all (assigned value 0) when divided by incremental powers of two (see example below).

For n = 2^k, all divisions give integers (not rounded at all) until n/2^(k+1), which rounds up to 1, and so a(2^k) = 1.

n/2^i shifts the bits of n down so the bit of n at position i-1 (least significant bit as position 0) is immediately below the radix point and so determines whether frac(n/2^i) >= 1/2 or < 1/2. frac(n/2^i) = 0 is when the bits of n at i-1 and below are all 0's. So a(n) is sum +1 for each 1-bit of n and -1 for each 0-bit of n but excluding any low 0's. - Kevin Ryde, Aug 31 2020

Examples

			For n = 10, a(10) = 0 + 1 + (-1) + 1 = 1.
		

Crossrefs

Cf. A000265, A007814, A145037.

Programs

JavaScript

var k = 1;
var r = 0;
for (var i = 0; i < 100; i += 1) {
    while ((i+1) >= Math.pow(2, k - 1)) {
        if (Math.round((i+1) / Math.pow(2, k)) < ((i+1) / Math.pow(2, k))) {
            r -= 1;
        } else if (Math.round((i+1) / Math.pow(2, k)) > ((i+1) / Math.pow(2, k))) {
            r += 1;
        } else {
            r += 0;
        }
        k += 1;
    }
    document.write(r, ", ");
    k = 1;
    r = 0;
}

Mathematica

Array[2 DigitCount[#, 2, 1] + IntegerExponent[#, 2] - Floor[Log2[#]] - 1 &, 80] (* Michael De Vlieger, Sep 01 2020 *)

PARI

a(n) = sum(k = 1, 1+logint(n, 2), my(x=(n % 2^k)/2^k); sign(round(x) - x)); \\ Michel Marcus, Aug 23 2020

PARI

a(n) = 2*hammingweight(n) + valuation(n,2) - logint(n,2) - 1; \\ Kevin Ryde, Aug 29 2020

Formula

a(n) = A145037(A000265(n)) = A145037(n) + A007814(n). - Kevin Ryde, Aug 31 2020