A139315 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.

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

Offset: 2

Author

J. Lowell, Jun 07 2008

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Ray Chandler, Table of n, a(n) for n = 2..100.

Crossrefs

Cf. A129902, A135060, A138511, A337686.

Extensions

a(14)-a(31) from Ray Chandler, Dec 03 2009

Name corrected by J. Lowell, Sep 14 2020

Name edited by Michel Marcus, Sep 15 2020