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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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) of this sequence is 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 other zeros like 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

LINKS

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

EXAMPLE

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.

CROSSREFS

Cf. A129902, A135060, A138511, A337686.

Sequence in context: A081125 A138570 A161887 * A014767 A002319 A195307

Adjacent sequences:  A139312 A139313 A139314 * A139316 A139317 A139318

KEYWORD

nonn

AUTHOR

J. Lowell, Jun 07 2008

EXTENSIONS

a(14)-a(100) from Ray Chandler, Dec 03 2009

Name corrected by J. Lowell, Sep 14 2020

Name edited by Michel Marcus, Sep 15 2020

STATUS

approved

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)