A377308 All winning positions of Gordon Hamilton's Jumping Frogs game, encoded as even numbers by their prime-factorization exponents.

2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 42, 48, 50, 54, 56, 60, 64, 70, 84, 90, 96, 100, 120, 126, 128, 140, 150, 162, 176, 192, 198, 200, 210, 240, 252, 256, 260, 264, 270, 280, 294, 300, 330, 350, 384, 390, 392, 400, 416, 420, 462, 480, 486, 490, 500

Offset: 1

Author

Glen Whitney, Oct 23 2024

Keywords

• nonn

Comments

For the rules of the Jumping Frogs game, see A377232.

Enumerate the primes in order, p_1 = 2, p_2 = 3, etc. Factor any natural number k > 1 as p_1^{x_1}p_2^{x_2}...p_i^{x_i}, where i is as small as possible and each x_j is nonnegative. Then when k is even and x_1, x_2, ..., x_i is a winning position for Jumping Frogs, k occurs as a term. We consider only even numbers to keep the positions distinct; leading zeros can never be used or affect the outcome of Jumping Frogs.

An even number k is a term if and only if A137502(k) is a term. - Pontus von Brömssen, Oct 24 2024

Examples

			Consider k = 28. It can be written as 2^2 * 3^0 * 5^0 * 7^1. The jumping frogs position 2, 0, 0, 1 has no legal moves (no occupied place adjacent to the 1 entry and no occupied place 2 places away from the 2 entry). Therefore it is not a winning position, and 28 is not a term.
Conversely, k = 20 can be written as 2^2 * 3^0 * 5^1. The jumping frogs position 2, 0, 1 can be won in a single move to 0, 0, 3 (all frogs in one place). Hence k is a term, namely a(8).
		

References

• See references at A377232.

Links

• Glen Whitney, Table of n, a(n) for n = 1..10000
• Glen Whitney, Python code that generated b-file

Crossrefs

Cf. A137502, A377232 (binary winning positions).