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Say that a number is an odd-bit number if the count of 1-bits in its binary representation is odd. Define an even-bit number analogously. Thus $541 = 1000011101_2$ is an odd-bit number, and $523 = 1000001011_2$ is an even-bit number.

Are there, asymptotically, as many odd-bit primes as even-bit primes?

For the first ten primes, we have $$\lbrace 10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101 \rbrace$$ with 1-bits $$\lbrace 1, 2, 2, 3, 3, 3, 2, 3, 4, 4 \rbrace$$ and so ratio $5/5=1$ at the 10-th prime. Here is a plot of this ratio up to $10^5$:

I would expect the ratio to approach $\frac{1}{2}$, except perhaps the fact that primes ($>2$) are odd might bias the ratio. The above plot does not suggest convergence by the 100,000-th prime (1,299,709).

Pardon the naïveness of my question.

Addendum: Extended the computation to the $10^6$-th prime (15,485,863), where it still remains 1.5% above $\frac{1}{2}.$\frac{1}{2}$: 5 remove dead links Say that a number is an odd-bit number if the count of 1-bits in its binary representation is odd. Define an even-bit number analogously. Thus$541 = 1000011101_2$is an odd-bit number, and$523 = 1000001011_2$is an even-bit number. Are there, asymptotically, as many odd-bit primes as even-bit primes? For the first ten primes, we have $$\lbrace 10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101 \rbrace$$ with 1-bits $$\lbrace 1, 2, 2, 3, 3, 3, 2, 3, 4, 4 \rbrace$$ and so ratio$5/5=1$at the 10-th prime.Here is a plot of this ratio up to$10^5$: I would expect the ratio to approach$\frac{1}{2}$, except perhaps the fact that primes ($>2$) are odd might bias the ratio. The above plot does not suggest convergence by the 100,000-th prime (1,299,709). Pardon the naïveness of my question. Addendum: Extended the computation to the$10^6$-th prime (15,485,863), where it still remains 1.5% above$\frac{1}{2}$: \frac{1}{2}.$

4 details

Say that a number is an odd-bit number if the count of 1-bits in its binary representation is odd. Define an even-bit number analogously. Thus $541 = 1000011101_2$ is an odd-bit number, and $523 = 1000001011_2$ is an even-bit number.

Are there, asymptotically, as many odd-bit primes as even-bit primes?

For the first ten primes, we have $$\lbrace 10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101 \rbrace$$ with 1-bits $$\lbrace 1, 2, 2, 3, 3, 3, 2, 3, 4, 4 \rbrace$$ and so ratio $5/5=1$ at the 10-th prime. Here is a plot of this ratio up to $10^5$:

I would expect the ratio to approach $\frac{1}{2}$, except perhaps the fact that primes ($>2$) are odd might bias the ratio. The above plot does not suggest convergence by the 100,000-th prime (1,299,709).

Pardon the naïveness of my question.

Addendum: Extended the computation to the $10^6$: 10^6$-th prime (15,485,863), where it still remains 1.5% above$\frac{1}{2}\$: