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A284156
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4-untouchable numbers.
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6
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298, 1006, 1016, 1108, 1204, 1502, 1940, 2370, 2770, 3358, 3482, 3596, 3688, 3976, 4076, 4164, 4354, 5206, 5634, 5770, 6104, 6206, 6696, 6850, 7008, 7118, 7290, 7496, 7586, 7654, 7812, 7922, 8494, 8546, 8786, 9108, 9122, 9326, 9340, 9476, 9836, 9952, 10318
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OFFSET
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1,1
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COMMENTS
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Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.
Let sk(n) denote the k-th iterate of s(n). 4-untouchable numbers are the numbers that lie in the image of s3(n), but not in the image of s4(n). Question: does the set of 4-untouchable numbers have a natural asymptotic density?
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LINKS
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EXAMPLE
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All even numbers less than 298 have a preimage under s4(n), so they are not 4-untouchable.
a(1) = 298, because 298 = s3(668) but 668 is untouchable. Therefore 298 is not in the image of s4(n). Note that 668 is the only preimage of 298 under s3(n).
a(2) = 1006, because 1006 = s3(5366) but 5366 is untouchable.
a(3) = 1016, because 1016 = s3(4402) = s3(5378) but both 4402 and 5378 are untouchable.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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