Количество синих чисел

Mapping out the rule

I'm thinking about coloring every pair of distinct natural numbers whose sum is prime in different colors, starting with 1 as red. This involves figuring out a coloring pattern for numbers 2 through 3000.

Coloring numbers

I’m painting numbers to ensure that every pair whose sum is prime has different colors. Since 1 is red, I’ll start by marking 2 blue.

Allocating colors

I'm thinking through colors for 5 and 6. Confirming that the sum of any two distinct numbers whose sum is prime, they have different colors from each other, and consistent with previous assignments.

Confirming the alignment

Checking the ongoing pattern: evens are blue and odds are red. For example, number 7 is red, fitting the established color scheme.

Focusing on patterns

I’m noticing that even numbers are blue and odd ones are red, except for 1, which is red. So far, this aligns with the rule about prime sums.

Noticing odd pairs

OK, let me see. Since the sum of two odd numbers is even and only 2 is an even prime, odd pairs don't share the same color under the "prime sum" rule.

Confirming color patterns

I’m thinking through the lack of restrictions for odd and even numbers sharing colors, as their sums can't be prime due to parity and distinctness. This suggests a clear pattern in coloring.

Confirming the pattern

It’s worth noting that this pattern aligns with the rule of prime sums and the established color assignments, showing no contradictions so far.

Confirming the tally

Interestingly, there are 1500 even and 1500 odd numbers between 1 and 3000, which supports the established coloring pattern by summing the numbers efficiently.

Thought about раскраски чисел и простых числе for 16 seconds