All-ones binary
The solid-color ceiling of binary — home of the Mersennes.
A number whose binary digits are all 1s, i.e. 2ᵏ−1 (a Mersenne number): 1, 3, 7, 15, 31… like 1023 = 1111111111₂.
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Only 19 all-ones binary numbers exist in the whole 0–1,000,000 range: 2¹−1 through 2¹⁹−1. The 20th, 2²⁰−1 = 1,048,575, is shut out by the range boundary — rarity by fence, not by luck.
Every all-ones binary number is automatically a binary palindrome (a run of 1s reads the same both ways), but the converse is far from true: only these 19 of the 2,000 binary palindromes are solid. The most extreme branch of the palindrome family.
These are the famous Mersenne numbers M(k) = 2ᵏ−1. Seven of the 19 in range are prime (3, 7, 31, 127, 8191, 131071, 524287, for k = 2, 3, 5, 7, 13, 17, 19) — nearly every largest-known prime has had this shape. Two millennia of prime hunting, and the hunt keeps returning to this street.