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A363369
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Number of steps x -> x+1 or x/prime required to go from n to 1.
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2
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0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 3, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 3, 2, 1, 3, 2, 3, 2, 2, 1, 3, 2, 3, 2, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 3, 3, 2, 1, 3, 2, 2, 2
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OFFSET
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1,4
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COMMENTS
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Each step is to any one of x+1 or x/p where p is a prime factor of x.
These steps are the game in A362416 and a(n) is the length of the shortest possible game starting at n.
a(n) = 1 iff n is a prime.
a(n) = 2 iff n is a semiprime or n+1 is a prime >= 5.
a(n) <= bigomega(n) since steps x/prime can divide out each prime successively (though often x+1 steps help make a shorter path to 1).
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LINKS
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FORMULA
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a(n) = Min_{i = n+1 or any n/prime} a(i), for n >= 2.
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EXAMPLE
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For n=80, steps 80 -> 40 -> 41 -> 1 (being /2, +1, /41) reach 1 after 3 steps and are the fewest possible so a(80) = 3.
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PROG
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(PARI) See links.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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