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A174457
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Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.
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2
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1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152
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OFFSET
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1,2
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COMMENTS
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In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.
Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.
a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022
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LINKS
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EXAMPLE
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9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.
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PROG
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(PARI) is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022
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CROSSREFS
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Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).
Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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