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

%I A377308 #16 Nov 15 2024 09:05:42
%S A377308 2,4,6,8,12,16,18,20,24,30,32,42,48,50,54,56,60,64,70,84,90,96,100,
%T A377308 120,126,128,140,150,162,176,192,198,200,210,240,252,256,260,264,270,
%U A377308 280,294,300,330,350,384,390,392,400,416,420,462,480,486,490,500
%N A377308 All winning positions of Gordon Hamilton's Jumping Frogs game, encoded as even numbers by their prime-factorization exponents.
%C A377308 For the rules of the Jumping Frogs game, see A377232.
%C A377308 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.
%C A377308 An even number k is a term if and only if A137502(k) is a term. - _Pontus von Brömssen_, Oct 24 2024
%D A377308 See references at A377232.
%H A377308 Glen Whitney, <a href="/A377308/b377308.txt">Table of n, a(n) for n = 1..10000</a>
%H A377308 Glen Whitney, <a href="/A377308/a377308.py.txt">Python code that generated b-file</a>
%e A377308 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.
%e A377308 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).
%Y A377308 Cf. A137502, A377232 (binary winning positions).
%K A377308 nonn
%O A377308 1,1
%A A377308 _Glen Whitney_, Oct 23 2024