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A365477
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The number of steps before reaching 1 or a repeated value, or -1 if neither occur, when starting at k = n and iterating k -> k/Lpf(k) - 1 if Omega(k) is even, else k -> k*Lpf(k) + 1, where Lpf(k) (A020639) is the least prime dividing k and Omega(k) (A001222) is the number of prime divisors of k counted with multiplicity.
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1
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0, 6, 3, 1, 5, 7, 90, 94, 7, 2, 99, 3, 105, 8, 2, 90, 93, 90, 90, 104, 8, 3, 11, 100, 2, 4, 91, 92, 96, 102, 114, 5, 3, 91, 8, 94, 90, 91, 4, 90, 103, 90, 94, 22, 100, 4, 96
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OFFSET
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1,2
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COMMENTS
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Of the starting values n so far studied, all iterative series with known outcomes either reach 1 or eventually fall into a 90-term loop whose lowest value is 7 - see the example for a(7) below. It is not known if other loops exist, nor whether an unbounded trajectory exists.
The first value whose outcome is unknown is a(48) - after 968 steps a number is reached which contains an unfactored divisor with 118 digits.
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LINKS
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Michael De Vlieger, Let c(n,k) be step k in the trajectory of n under the recursion described in the Name. Plot c(n,k) at (x,y) = (n,k), n = 1..240, 1 <= k <= 128, 8X magnification, with a color function showing 1 in black, primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Shows a certain tendency of the class of c(n,k) (prime, squarefree composite, etc.) to recur at c(n+1,k+2), and a different behavior for n in {48, 73, 97, 98, 99, ...} devoid of prime powers for larger k.
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EXAMPLE
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a(2) = 6. The series of steps to reach 1 is : 2, 5, 26, 12, 25, 4, 1, taking six steps in total.
a(5) = 5. The series of steps to reach 1 is : 5, 26, 12, 25, 4, 1, taking five steps in total.
a(7) = 90. The series of steps before 7 reappears is : 7, 50, 101, 10202, 5100, 2549, 6497402, 3248700, 6497401, 2548, 5097, 1698, 3397, 78, 157, 24650, 49301, 7042, 14085, 4694, 2346, 1172, 2345, 11726, 5862, 11725, 2344, 1171, 1371242, 685620, 1371241, 1170, 2341, 5480282, 2740140, 5480281, 2340, 1169, 166, 82, 40, 19, 362, 180, 361, 18, 37, 1370, 2741, 7513082, 15026165, 75130826, 37565412, 18782705, 93913526, 187827053, 4368070, 2184034, 1092016, 546007, 78000, 156001, 2136, 4273, 18258530, 9129264, 18258529, 4272, 2135, 10676, 5337, 16012, 32025, 96076, 192153, 64050, 32024, 16011, 5336, 10673, 820, 409, 167282, 83640, 167281, 408, 817, 42, 85, 16, 7, taking ninety steps in total.
a(8) = 94. The series of steps before repeating the value 82 begins 8, 17, 290, 581, 82, which now enters the same iterative loop of numbers as a(7) and thus repeats 82 after 90 more steps, so a(8) = 4 + 90 = 94.
a(176) = 955. See the attached text file. The maximum value in this series, which reaches 1 after 955 steps, is the 75 digit number 30007...78878.
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MATHEMATICA
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With[{k = 2^7}, Array[If[# >= k, -1, #] &@ LengthWhile[NestWhileList[If[EvenQ@ PrimeOmega[#1], #1/#2 - 1, #1 #2 + 1] & @@ {#, FactorInteger[#][[1, 1]]} &, #, Unequal, All, k, -1], # > 1 &] &, 120] ] (* Set k to run the recursion up to k times; terms that iterate k times are deemed -1, Michael De Vlieger, Sep 15 2023 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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