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A328520
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GCD of terms in A002182 that have n prime factors counted with multiplicity.
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1
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1, 2, 2, 12, 12, 12, 60, 120, 2520, 2520, 2520, 55440, 55440, 720720, 720720, 12252240, 36756720, 698377680, 3491888400, 80313433200, 160626866400, 160626866400, 9316358251200, 288807105787200, 2021649740510400, 74801040398884800, 74801040398884800, 3066842656354276800, 131874234223233902400
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OFFSET
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0,2
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COMMENTS
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If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t (Cf. A328523) then (1): a(n+1) is a multiple of a(n) for n >= 1 and (2): this sequence can be used to find terms to A199337 and A330737.
Proof of (1): Let Generation(n) be the terms in A002182 with n prime factors counted with multiplicity. Then a(n) = GCD(Generation(n)). As each term in Generation(n) is of the form a(n) * t for some t and each term in Generation(n + 1) is p * g for some g in Generation(n), a(n + 1) is a multiple of a(n).
Proof of (2): As a(n + 1) is a multiple of a(n) we have that if m | a(n), we also have m | a(n + k), k >= 0 hence the largest number not divisible by m can have at most n - 1 prime factors counted with multiplicity.
Given Generation(n) we can find all candidates for Generation(n + 1) and from there on find terms to A002182, A199337 and A330737.
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LINKS
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EXAMPLE
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The terms in A002182 with n = 4 prime divisors counted with multiplicity are 24, 36 and 60. Their GCD is 12 hence a(4) = 12.
Furthermore, If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t then we have that each term with more than 4 prime divisors counted with multiplicity is a multiple of at least one of 24, 36 or 60 hence is divisible by 12.
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MATHEMATICA
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(* First, load the function f at A025487, then: *)
Block[{s = Union@ Flatten@ f@ 20, t}, t = DivisorSigma[0, s]; s = Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]; t = PrimeOmega[s]; Drop[Array[GCD @@ s[[Position[t, #][[All, 1]] ]] &, Max@ t + 1, 0], -3] ] (* Michael De Vlieger, Jan 12 2020 *)
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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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