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%I #34 Mar 17 2022 01:05:44
%S 1,2,6,12,60,120,1260,840,0,2520,27720,55440,0,720720,1081080,2162160,
%T 61261200,36756720,1396755360,2327925600,0,698377680,16062686640,
%U 48188059920,0,749592043200,160626866400,240940299600,0,6987268688400
%N 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.
%C 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.
%C Generalizing above result, a(pq)=0 for distinct primes p,q with p < q if p^2 < q. - _Ray Chandler_, Dec 03 2009
%C 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
%C a(n) is the least k such that A337686(k) = n, or 0 if there is no such k. - _Michel Marcus_, Sep 16 2020
%H Ray Chandler, <a href="/A139315/b139315.txt">Table of n, a(n) for n = 2..100</a>.
%e 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.
%Y Cf. A129902, A135060, A138511, A337686.
%K nonn
%O 2,2
%A _J. Lowell_, Jun 07 2008
%E a(14)-a(31) from _Ray Chandler_, Dec 03 2009
%E Name corrected by _J. Lowell_, Sep 14 2020
%E Name edited by _Michel Marcus_, Sep 15 2020
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