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For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of ...
[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …] [OEIS 100] Computed up to 1 011 597 392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property. [Mw 85]
The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities ρ 1, ρ 2, ρ 3, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the ...
This Dutch oven comes in a wide range of sizes, from 2 quarts all the way up to 13.25 quarts, and we tested the popular 7.25-quart size, which is recommended for cooking seven or eight servings at ...
Simply put, each number equals a triangular number plus 1. These are the first number on each row of Floyd's triangle. The lazy caterer's sequence (green) and other OEIS sequences in Bernoulli's triangle. As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ...
1 1/4 tsp. kosher salt, divided, plus more to taste. ... 1 lb. 90/10 lean ground beef. 3 large garlic cloves, finely chopped (about 1 Tbsp.) ... 4 cups chicken broth. 2 cups water. 1 Tbsp ...
The last step uses the fact that p 2 divides 2 p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2 p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2 p(p−1) = p 2 (...) + 1. And now with a similar calculation as above, the following results:
Plus, I'll reveal the answers further down. Related: The 26 Funniest NYT Connections Game Memes You'll Appreciate if You Do This Daily Word Puzzle Hints About Today's NYT Connections Categories on ...