Barely Abundant Numbers

Today's diurnal age of 26742 days threw up an interesting number property that I can't recall coming across before. This is not surprising because the last occurrence occurred when I was 17816 days old, long before I started keeping track of the numbers associated with my diurnal age. The property in question qualifies both numbers for inclusion in OEIS A071927:

A071927

Barely abundant numbers: abundant \(n\) such that \( \dfrac{\sigma(n)}{n }< \dfrac{\sigma(m)}{m} \)for all abundant numbers \(m<n,\) \( \sigma(n) \) being the sum of the divisors of \(n\).

The terms in the sequence up to one million are:

12, 18, 20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 26742, 26778, 26886, 26898, 26958, 27042, 27078, 27102, 27114, 27138, 27282, 27294, 27366, 27402, 27498, 27546, 27582, 27618, 27726, 27822, 27834, 27858, 27894, 27906, 27942, 27978, 28038, 28074, 28146, 28218, 28326, 28338, 28374, 28398, 28506, 28554, 28698, 28722, 28734, 28758, 28794, 28806, 28878, 28902, 28986, 29166, 29226, 29262, 29334, 29418, 29454, 29514, 29586, 29598, 29622, 29658, 29706, 29742, 29802, 29814, 29838, 29922, 29958, 29994, 30018, 30054, 30066, 30126, 30138, 30234, 30306, 30354, 30462, 30486, 30522, 30594, 30606, 30642, 30678, 30714, 30882, 30918, 31002, 31026, 31074, 31134, 31182, 31254, 31362, 31386, 31398, 31422, 31566, 31638, 31674, 31686, 31782, 31818, 31854, 31938, 31998, 32082, 32106, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752

Figure 1 shows a plot of these values using a vertical log axis and the long evenly spaced stretch from 26742 to 32128 stands out clearly.

Figure 1: permalink

Figure 2 shows the same numbers showing the sigma(n)/n ratios and the factorisation (click on the image to enlarge):

Figure 2: permalink

It can be seen that all the barely abundant numbers from 26742 to 32128 are sphenic numbers of the form 2 x 3 x prime. Interesting all these numbers are admirable numbers, that is numbers whose proper factors add to the number when one of the factors is made negative. With these numbers the factor to be made negative is always 6. For example, the factors 1, 2, 3, -6, 4457, 8914, 13371  add to the admirable number 26742 and the factors 1, 2, 3, -6, 5039, 10078, 15117 add to the admirable number 30234.

It's easy to see why the 2 x 3 x prime are so successful in forming barely abundant numbers. The divisors of a number \(n\) in that case are \(2, 3, n/6, n/3, n/2, n\) and the sum of these divisors is \(2n+5\), giving a \( \sigma(n)/n \) ratio of \(2+5/n \). As \(n\) gets larger, the ratio gets smaller as can be seen in the progressive decrease in the size of the ratio from 26742 to 32128. Why the abrupt gap occurs after 32128 I'm not sure.

So barely abundant numbers will be cropping up quite regularly for the next six to seven years until another "drought" of such numbers occurs between 32129 and 77743.