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A139315
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a(n) is the smallest integer k such that n*k is the smallest multiple of k with twice as many divisors as k, or 0 if no such number is possible.
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6
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1, 2, 6, 12, 60, 120, 1260, 840, 0, 2520, 27720, 55440, 0, 720720, 1081080, 2162160, 61261200, 36756720, 1396755360, 2327925600, 0, 698377680, 16062686640, 48188059920, 0, 749592043200, 160626866400, 240940299600, 0, 6987268688400
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OFFSET
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2,2
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COMMENTS
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Proof that a(10)=0: In order for 10*k to have twice as many divisors as k, it must be either a multiple of 20 but not of 40 or 100 (in which case 8*k has twice as many divisors) or a multiple of 50 but not of 100 or 250 (in which case 4*k has twice as many divisors). In both cases, 10*k is not the smallest number with twice as many divisors as k and so a(10)=0.
Generalizing above result, a(pq)=0 for distinct primes p,q with p < q if p^2 < q. - Ray Chandler, Dec 03 2009
That is, a(m)=0 for m in A138511, but there are also other zeros, such as those at n = 30, 50, 68, 76, 90, 92, 98, ... - Michel Marcus, Sep 14 2020
a(n) is the least k such that A337686(k) = n, or 0 if there is no such k. - Michel Marcus, Sep 16 2020
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LINKS
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EXAMPLE
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a(8) = 1260 because it must be a multiple of 4 but not of 8. It cannot be 4 because 4*3=12 has twice as many divisors as 4. It cannot be 12 because 12*5=60 has twice as many divisors as 12. It cannot be 60 because 60*6=360 has twice as many divisors as 60. It cannot be 180 because 180*7=1260 has twice as many divisors as 180. It must be 1260.
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CROSSREFS
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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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