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A327406
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Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).
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5
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0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2
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OFFSET
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1,15
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers whose prime indices have a common divisor > 1 are listed in A318978.
Note that A318978 includes also all odd primes and their powers, thus the only numbers for which a maximum such divisor is 1 are the powers of 2. Therefore A000079 gives the indices of zeros in this sequence. - Antti Karttunen, Dec 06 2021
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LINKS
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EXAMPLE
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We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.
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MATHEMATICA
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Table[Length[FixedPointList[#/Max[Select[Divisors[#], GCD@@PrimePi/@First/@FactorInteger[#]!=1&]]&, n]]-2, {n, 100}]
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PROG
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(PARI)
A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi, factor(d)[, 1]~))>1), divisors(n))));
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CROSSREFS
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First appearance of n is A080696(n).
See link for additional cross-references.
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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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