Fibonacci Numbers in the Abundancy Index

I've made several posts over the years concerning numbers and their associated abundancy. The abundancy of a number \(n\) is defined as:$$ \frac{\sigma_1(n)}{n} $$The abundancy of a number is sometimes referred to as its abundancy index. The number associated with my diurnal age today, 27280, is a member of OEIS  A349687:

A349687

Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

The initial members of the sequence are (permalink):

1, 2, 6, 15, 24, 26, 28, 84, 90, 96, 120, 270, 330, 496, 672, 1335, 1488, 1540, 1638, 8128, 24384, 27280, 44109, 68200, 131040, 447040, 523776, 18506880, 22256640, 33550336, 36197280, 38257095, 65688320, 91963648, 95472000, 100651008, 102136320, 176432256, 197308800

The initial Fibonacci numbers are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657

In the case of 27280 we have:$$ \begin{align} \frac{\sigma_1(27280)}{27280} &= \frac{71424}{27280}\\ &= \frac{144}{55} \end{align} $$We find that two earlier members of the OEIS sequence, 330 and 1540, have this same abundancy as do two later members, 68200 and 447040. I only checked up to one million so there will be many more numbers with the same abundancy as 27280. Numbers with the same abundancy are called friendly numbers. These number properties are base independent.

A variation on the above would be use the set of square numbers instead of the Fibonacci numbers. Up to 40,000, the square numbers are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 11449, 11664, 11881, 12100, 12321, 12544, 12769, 12996, 13225, 13456, 13689, 13924, 14161, 14400, 14641, 14884, 15129, 15376, 15625, 15876, 16129, 16384, 16641, 16900, 17161, 17424, 17689, 17956, 18225, 18496, 18769, 19044, 19321, 19600, 19881, 20164, 20449, 20736, 21025, 21316, 21609, 21904, 22201, 22500, 22801, 23104, 23409, 23716, 24025, 24336, 24649, 24964, 25281, 25600, 25921, 26244, 26569, 26896, 27225, 27556, 27889, 28224, 28561, 28900, 29241, 29584, 29929, 30276, 30625, 30976, 31329, 31684, 32041, 32400, 32761, 33124, 33489, 33856, 34225, 34596, 34969, 35344, 35721, 36100, 36481, 36864, 37249, 37636, 38025, 38416, 38809, 39204, 39601, 40000

We find only 18 numbers qualify (permalink): 

1, 40, 81, 135, 216, 224, 400, 819, 1372, 3240, 3744, 4650, 6318, 18144, 21700, 27930, 30240, 32760

Here is the breakdown:

1 --> 1/1 = 1/1
40 --> 90/40 = 9/4
81 --> 121/81 = 121/81
135 --> 240/135 = 16/9
216 --> 600/216 = 25/9
224 --> 504/224 = 9/4
400 --> 961/400 = 961/400
819 --> 1456/819 = 16/9
1372 --> 2800/1372 = 100/49
3240 --> 10890/3240 = 121/36
3744 --> 11466/3744 = 49/16
4650 --> 11904/4650 = 64/25
6318 --> 15288/6318 = 196/81
18144 --> 60984/18144 = 121/36
21700 --> 55552/21700 = 64/25
27930 --> 82080/27930 = 144/49
30240 --> 120960/30240 = 4/1
32760 --> 131040/32760 = 4/1

These numbers belong to OEIS A069070.

Highly Composite Deficient Numbers

My diurnal age today is 27027 and this factorises as follows:$$27027=3^3 \times 7 \times 11 \times 13$$Although this number, with its many factors and 32 divisors, looks as though it should be abundant, it's not. It just misses the mark because the ratio of the sum of its proper divisors to the number itself just falls short of unity:$$ \begin{align} \frac{ \sigma(27027, 1)-27027}{27027}&=\frac{53760-27027}{27027}\\ &=\frac{26733}{27027} \\ & \approx 0.989121989121989 \end{align}$$On March 24th 2023, I wrote about Balanced Numbers and 27027 is such a number because:$$27027=\overbrace{27}^{2+7=9} \cdot 0 \cdot \overbrace{27}^{2+7=9}$$However, 27027 has a greater claim to fame because it's a member of OEIS A302934:

A302934

Highly composite deficient numbers: deficient numbers \(k\) whose number of divisors \(d(k) \gt d(m) \) for all deficient numbers \(m \lt k\).

The table below shows a list of deficient numbers up to one million that have a record number of divisors. The ratio of the sum of proper divisors to the number is also shown (permalink).

 number   divisors   ratio

  1        1          0.000000000000000
  2        2          0.500000000000000
  4        3          0.750000000000000
  8        4          0.875000000000000
  16       5          0.937500000000000
  32       6          0.968750000000000
  64       7          0.984375000000000
  105      8          0.828571428571429
  225      9          0.791111111111111
  315      12         0.980952380952381
  1155     16         0.994805194805195
  2475     18         0.953939393939394
  4455     20         0.955555555555556
  8775     24         0.978347578347578
  26325    30         0.994833808167142
  27027    32         0.989121989121989
  63063    36         0.974025974025974
  106029   40         0.971988795518207
  247401   48         0.990614427589217
  693693   54         0.988980716253444
  829521   60         0.995464852607710
  969969   64         0.995280261534132

Looking at the table, the status of 27027 as a record breaker can be seen. Deficient numbers can be ranked by their number of divisors or by how close they approach unity (or how close they approach 2 if we prefer to deal with abundancy). I've investigated the latter in a post titled Odd Deficient Numbers from April 30th 2021. Another post on deficient numbers is Gaps Between Deficient Numbers from October 30th 2020. The post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019 is also relevant.

Odd Deficient Numbers

I've written about deficient numbers in an eponymous post on January 28th 2018 and in which I mentioned, for the first time in my postings, the ratio between the sum of the divisors of a number and the number itself viz. \( \displaystyle \frac{\sigma_1(n)}{n} \).

In that post, I didn't refer to the ratio by its name of abundancy but in later posts I explored the concept of abundancy in more detail. Here are links to posts in which it was discussed:

• Multiperfect, Hyperperfect and Superperfect Numbers on July 24th 2019
• Friendly versus Solitary Numbers on December 16th 2020
• Hemiperfect Numbers on January 3rd 2021

The last two posts, as can be noted, are relatively recent. Today, in turning 26325 days old, I encountered a reference to abundancy once again. Specifically in the context of OEIS A188597:

A188597

Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

My diurnal age is a member of this sequence which runs:

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

 Figure 1 shows the progression:

Figure 1

Not surprisingly, all these numbers are highly composite, despite all being deficient. This can be seen in Figure 2 where a table of factorisation and number of divisors is presented.

Figure 2

There are a number of related sequences, one of them is OEIS A171929. 

A171929

Odd numbers whose abundancy is closer to 2 than any smaller odd number.

Here the numbers need only to odd and can be abundant or deficient. Figure 3 shows the abundancy and its absolute difference from 2.

Figure 3

Another related sequence is OEIS A188263:

A188263

Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

 

In this sequence, all the numbers are abundant and thus their abundancy is greater than 2. Figure 4 shows a table of the initial numbers and their abundancies.

Figure 4

It's interesting to consider the idea that the limit of the abundancy of these sorts of odd abundant and odd deficient numbers is actually 2 as their abundancy can be as close to 2 as desired. 

Hemiperfect Numbers

The so-called hemiperfect numbers relate a concept called abundancy that I've dealt with in two previous posts:

• Multiperfect, Hyperperfect and Superperfect Numbers (July 24th 2019) 
• Friendly versus Solitary Numbers (December 16th 2020)

I'll define the abundancy of a number \(n\) once again to be the ratio of the sum-of-divisors of \(n\) to \(n\) itself. It is given by the formula:$$ \frac{\sigma_1(n)}{n} \text{ where }\sigma_1(n) \text{ is the divisor function}$$Note that abundancy may also be defined as:$$\sigma_{-1}(n) \text{ where } \sigma_{-1}(n) \text{ represents the sum of the reciprocals of the divisors of } n$$A multiperfect (sometimes multiply perfect) number is a number whose abundancy is a whole number:$$\frac{\sigma_1(n)}{n}=k \text{ with } k \text{ an integer } \geq 2$$We can refer to such a number as \(k\)-perfect with 2-perfect numbers being the perfect numbers 6, 28, 496, 8128 etc.

Today I turned 26208 days old and discovered that this number is a member of OEIS A159907:

A159907

Numbers \(n\) with half-integral abundancy index such that:                  $$\frac{\sigma_1(n)} {n} = k+\frac{1}{2} \text{ with integer }k$$

Numbers of this sort are termed hemiperfect. The sequence, up to 26208, consists of 2, 24, 4320, 4680, 26208 where:

$$\begin{align}
\frac{\sigma_1(2)} {2} &= \frac{3}{2}=1+\frac{1}{2}\\
\frac{\sigma_1(24)} {24} &= \frac{60}{24}=2+\frac{1}{2} \\
\frac{\sigma_1(4320)} {4320} &= \frac{15120}{4320}=3+\frac{1}{2} \\
\frac{\sigma_1(4680)} {4680} &= \frac{16380}{4680}=3+\frac{1}{2} \\
\frac{\sigma_1(26208)} {26208} &= \frac{91728}{26208}=3+\frac{1}{2}
\end{align}$$After this the numbers get bigger quickly. The next hemiperfect number is 8910720 which has an abundancy of 9/2 and is termed 9-hemiperfect. Similarly, 2 is 3-hemiperfect, 24 is 5-hemiperfect and 4320, 4680 and 26208 are 7-hemiperfect.

Friendly versus Solitary Numbers

Today I turned 26190 days old and discovered that this number forms one half of a friendly pair of numbers. The other half is 8148. What do these two numbers have in common? Well, we find that:$$ \frac{\sigma_1(26190)}{26190}=\frac{70560}{26190}=\frac{784}{291} \text{ and } \frac{\sigma_1(8148)}{8148}=\frac{21952}{8148}=\frac{784}{291}$$So if the sum of the divisors of one number divided by that number is the same as the sum of the divisors of another divided by that other number, then the numbers are said to be friendly. Friendly numbers are not to be confused with amicable numbers where the numbers are related in such a way that the sum of the proper divisors of one is equal to the sum of the proper divisors of the other. The smallest pair of amicable numbers is 220 and 284. 

Getting back to friendly numbers, we find that friendly triples and higher-order tuples are also possible. Friendly triples include: 

• (2160, 5400, 13104)
• (9360, 21600, 23400)
• (4320, 4680, 26208)

Friendly quadruples include: 

• (6, 28, 496, 8128)
• (3612, 11610, 63984, 70434)
• (3948, 12690, 69936, 76986)

Friendly quintuples include:

• (84, 270, 1488, 1638, 24384)
• (30, 140, 2480, 6200, 40640)
• (420, 7440, 8190, 18600, 121920)

Numbers that have friends are called friendly numbers, and numbers that do not have friends are called solitary numbers.

This ratio of the sum-of-divisors of an integer \(n\) to the integer itself is termed its abundancy and is defined as: \( \displaystyle \frac{\sigma_1(n)}{n}\).

By this definition, two numbers are friendly is they have the same abundancy.  Two numbers with the same abundancy form a friendly pair; \(n\) numbers with the same abundancy form a friendly \(n\)-tuple. 

Abundancy may also be expressed as \( \sigma _{-1}(n)\) where \( \sigma _{k} \) denotes the sum of the \(k\)-th powers of the divisors of \(n\). When \(k\)=-1, we have the sum of the reciprocals of the divisors. The abundancy of a number \(n\) should not be confused with its abundance \( A(n) \equiv  \sigma_1(n)-2n \). Refer to WolframMathWorld.

From Wikipedia we learn that:

if the numbers \(n\) and \( \sigma(n) \) are coprime – meaning that the greatest common divisor of these numbers is 1, so that \( \sigma(n)/n \) is an irreducible fraction – then the number \(n\) is solitary. For a prime number \(p\), we have \( \sigma_1(p) = p + 1\), which is co-prime with \(p\).

Thus all primes and multiples of primes are solitary. Wikipedia continues:

No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least \(10^{30}\). Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.

Mutually friendly numbers as we said earlier can form friendly \(n\)-tuples that might be considered families or clubs. It's an open question whether these families have an infinite number of members. For example, it is conjectured that there are infinitely many perfect numbers but only 51 are currently known. Each perfect number has an abundancy of 2 and thus currently the perfect numbers form a 51-tuple or a family with 51 members. 

Similarly multiply perfect numbers form friendly families but firstly let's define what is meant by a multiply perfect numbers:

For a given natural number \(k\), a number \(n\) is called \(k\)-perfect (or \(k\)-fold perfect) if and only if the sum of all positive divisors of \(n\) (the divisor function, \( \sigma(n) \), is equal to \(k \times n\); a number is thus perfect if and only if it is 2-perfect. A number that is \(k\)-perfect for a certain \(k\) is called a multiply perfect number. As of 2014, \(k\)-perfect numbers are known for each value of \(k\) up to 11. Source. Also see my blog post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019.

The club of friendly numbers with abundancy equal to 9 has 2094 known members but these multiply perfect clubs or families are thought to be finite (unlike the perfect family that is conjectured to be infinite).

There are a number of OEIS sequences associated with friendly and solitary numbers. It was stated earlier that numbers that are coprime with their sum of divisors are solitary but this is sufficient and not necessary condition for solitariness. OEIS A095739 lists those numbers that are solitary and yet not coprime with their sum of divisors:

A095739

Numbers known to be solitary but not coprime to sigma.

The first of these numbers are 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369, ...

26190, the number that began this post, is a member of OEIS A050973:

A050973

Larger member of friendly pairs ordered by smallest maximal element.

The initial member of this sequence are:

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1488, 1540, 1638, 1638, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, ...

The smaller members of these pairs are given by OEIS A050972:

A050972

Smaller member of friendly pairs ordered by smallest maximal element.

The initial members of this sequence are:

6, 30, 80, 40, 12, 84, 66, 78, 102, 6, 28, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 84, 270, 330, 84, 270, 1488, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 30, 140, 534, 132, 1040, 570, 582, 606, ...

From these sequences, we can form the various pairs e.g. 28 and 6, 140 and 30 etc. Notice the two numbers (819 and 135) marked in bold in the above sequences. This pair are an example of two odd numbers being friendly. There are also cases of even being friendly to odd, such as 42 and 544635 with abundancy 16/7.