The Antisigma Function

The \( \textbf{antisigma}\) function returns the sum of the proper non-divisors of \(n\). I hadn't heard of it before but came across reference to it when researching properties of the number 77. Let's designate this function as a(\(n\)) and use 15 as an example to illustrate how it works. The proper divisors of 15 are 1, 3 and 5. This means that 2, 4, 6, 7, 8, 9, 10, 11, 12, 13 and 14 are non-divisors and they total 96. Thus:$$a(15)=96$$A quicker way to calculate the sum of proper non-divisors is to use the following formula that makes use of the sum of the terms of an arithmetic sequence:$$ \begin{align} \text{a}(n) &= \frac{n(n+1)}{2} - \sigma(n) \\ \text{a}(15) &= \frac{15 \times 16}{ 2} - 24 \\ &=140-24\\&=96 \end{align} $$I had to be reminded as to the formula for the sum of the terms of an arithmetic progression with starting term \(a\) and common difference \(d\). The sum \( \text{S}_n \) of the first \(n\) terms is given by:$$ \begin{align} \text{S}_n &= \frac{n}{2}(2a+(n-1)d) \\ &= \frac{n (n+1)}{2} \text{ for }a=1 \text{ and } d=1 \end{align}$$The antisigma function differs from Euler's totient function that counts the number of integers up to a given number \(n\) that are coprime to \(n\). In the case of 15, we have:$$ \begin{align} \phi(15) &= 15 \times (1-\frac{1}{3}) \times (1-\frac{1}{5}) \\ &= 15 \times \frac{2}{3} \times \frac{4}{5} \\ &=8 \end{align}$$The eight numbers that are coprime to 15 are 1, 2, 4, 7, 8, 11, 13 and 14. 

The antisigma function relates to the non-divisors of a number and these differ from its antidivisors. The antidivisors of 15 are 2, 6 and 10. See blog More on Anti-divisors.

The OEIS includes various sequences relating to antisigma function. First and foremost there is OEIS A024816:

A024816: antisigma(\(n\)) which is the sum of the numbers less than \(n\) that do not divide \(n\).

This sequence begins:

0, 0, 2, 3, 9, 9, 20, 21, 32, 37, 54, 50, 77, 81, 96, 105, 135, 132, 170, 168, 199, 217, 252, 240, 294, 309, 338, 350, 405, 393, 464, 465, 513, 541, 582, 575, 665, 681, 724, 730, 819, 807, 902, 906, 957, 1009, 1080, 1052, 1168, 1182, 1254, 1280, 1377, 1365

Then there is OEIS A200981:

A200981: numbers \(k\) such that the sum of non-divisors of \(k\) is prime.

3, 4, 10, 21, 34, 46, 58, 70, 85, 93, 118, 129, 130, 144, 178, 201, 226, 237, 262, 298, 310, 322, 324, 325, 333, 334, 346, 382, 406, 418, 430, 466, 478, 502, 513, 514, 517, 549, 598, 622, 633, 634, 657, 658, 669, 706, 730, 742, 813, 826, 837, 838, 865, 922, 982, 985

77 Sunset Strip

With my 77th birthday approaching, I thought I'd prepare by finding interesting facts about the number. The Sunset Strip reference is to Bingo Calls and recalls the famous television series of the sixties called 77 Sunset Strip that I used to watch in my youth.

So what are some of the interesting mathematical properties of the number 77. Let's start with partitions.

• 77 is a repdigit, a number whose digits are all equal (permalink).
  
  
• 77 is a semiprime and the product of two successive primes, 7 and 11. It is the 22nd discrete (or squarefree) semiprime.
  
  
• 77 is a partition number, specifically the number of partitions into which 12 can be divided e.g. [5, 4, 3]. I won't list them all here as it looks a bit messy but here is a permalink. The partition numbers form OEIS A000041.
  
  
• 77 is the number of partitions of 28 into prime parts e.g. [5, 23] or [2, 7, 19].
  
  
• 77 is the sum of the first eight primes: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77. These progressive sums of the primes form OEIS A007504:
  
  0, 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888, ... 
  
  
• 77 is the smallest integer requiring five syllables (sev-en-ty-sev-en) when written in words.
  
  
• 77 is a member of OEIS A056964: a(\(n\)) = \(n\) + reversal of digits of \(n\). Here \(n=16\) and \(16 + 61 = 77\).
  
  
• 77 is the sum of three consecutive squares:$$4^2 +5^2 +6^2 = 77$$
• 77 can be generated in only way from the rule \(n+\text{ SOD}(n)\) when \(n=70\) since \(70+7=77\). Thus it is not a self number nor a junction number.
  
  
• 77 is a member of OEIS A004207: a(0) = 1, a(\(n\)) = sum of digits of all previous terms. The initial terms are 1, 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77.
  
  
• 77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1. See Appendix for a discussion of this property (using Gemini).
  
  
• 77 and 49 are the only 2-digit numbers whose home primes (in base 10) have \( \textbf{not}\) been calculated.
  
  
• 77 is a Blum integer since both 7 and 11 are Gaussian primes. A natural number \(n\) is a Blum integer if \(n = p × q\) is a semiprime for which \(p\) and \(q\) are distinct prime numbers congruent to 3 mod 4. The first few Blum integers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ...
  
  
• 77 has a prime aliquot sum of 19 leading to an aliquot sequence of 77, 19, 1, 0. It is the second composite member of the 19-aliquot tree with 65 being the other member.
  
  
• 77 is the value returned by the antisigma function when it is applied to 13. Thus 77 is the sum of the proper non-divisors of 13 which, because 13 is prime, includes all the integers from 2 to 12. Using the formula we have:$$ \begin{align} \text{a}(n) &=\frac{n(n+1)}{2}-\sigma(n)\\  \text{a}(13) &= \frac{13 \times 14}{2}-14 \\ &=91-14 \\ &=77 \end{align}$$

I'll keep adding to this as I discover new properties

Appendix (Gemini Prompt and Response):

Why is 77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1. 

The Phenomenon

To understand why 77 is the final exception, we first have to look at what it means for a number to have this property. We are looking for a set of distinct positive integers that add up to a specific target number, while their reciprocals add up to exactly 1.

For example, 11 works perfectly:

• The numbers are 2, 3, and 6.

• Their sum is 2 + 3 + 6 = 11.

• Their reciprocals are 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 1.

Another example is 24, which can be formed by the set 2, 4, 6, and 12 (their sum is 24, and 1/2 + 1/4 + 1/6 + 1/12 = 1).

As the target number gets larger, you have more "room" to include more numbers. This opens up vastly more combinatorial possibilities for finding a set that works.

The Combinatorial Tipping Point

The reason 77 is the largest number that cannot be written this way is not due to a single geometric rule or elegant equation. Instead, 77 represents the threshold of a combinatorial tipping point.

In 1963, mathematician Ronald Graham published a paper titled "A Theorem on Partitions." He proved that beyond 77, the number of possible integer combinations becomes dense enough that every single integer to infinity can be successfully partitioned this way. 77 is simply the final "gap" before the overlapping possibilities cover everything.

Graham's Proof: The "Hopping" Mechanism

To prove that every number above 77 works, Graham didn't test numbers to infinity. Instead, he found a mathematical mechanism to use small known solutions to build infinitely larger ones.

Here is how his mechanism works:

1. Halving the Reciprocals: Imagine you have a valid set of numbers that sums to a target, let's call it N. If you multiply every number in that set by 2, the total sum doubles to 2N. However, the sum of their reciprocals is cut exactly in half, dropping from 1 to 1/2.

2. Filling the Gap: To get the reciprocals back up to 1, you just need to add a new set of distinct numbers to your pool whose reciprocals sum to exactly 1/2.

3. Creating the "Hops": There are many known sets whose reciprocals sum to 1/2. For example:

   • The set {3, 6} works (1/3 + 1/6 = 1/2). The sum of these two numbers is 9. So, from your original N, you have just generated a new valid set summing to 2N + 9.

   • The set {3, 7, 78, 91} works (1/3 + 1/7 + 1/78 + 1/91 = 1/2). The sum of these numbers is 179. So, using this set, you can generate a new valid set summing to 2N + 179.

By discovering several of these "hopping" formulas (2N + a, 2N + b, 2N + c), Graham showed that if you could just find a continuous block of valid numbers to start with as your base, the formulas would generate overlapping sequences that hit every possible integer from that block onward forever.

Why Exactly 77?

Graham manually mapped out the base cases, proving that starting at 78, the chain reaction of these formulas takes over and leaves no gaps to infinity. For instance, he demonstrated that 78 can be perfectly partitioned into 2, 6, 8, 10, 12, and 40 (which sum to 78, and 1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1).

But what about the numbers below 78?

Mathematician D.H. Lehmer analyzed the finite number of combinations available for smaller sums. For a sum to be relatively small, you are restricted to using a small amount of small integers. However, small integers have large reciprocals (like 1/2 or 1/3), which makes it incredibly difficult to combine them to hit exactly 1 without overshooting it.

Through exhaustive computational search and bounding arguments, Lehmer proved that no valid combination of integers summing to 77 exists. Because Graham's inductive chain reaction successfully covers absolutely everything from 78 upward, 77 is left stranded as the absolute largest exception to the rule.

So what numbers below 77 can be written as sum of reciprocals adding to 1.

In mathematics, integers that can be formed by the sum of a set of distinct positive integers whose reciprocals add up exactly to 1 are known as Strict Egyptian numbers. The name is a nod to "Egyptian fractions," which are sums of distinct unit fractions (like 1/2 + 1/3 + 1/6).

While every single integer from 78 onward to infinity is a Strict Egyptian number, the numbers below 78 are riddled with gaps. This happens because, for smaller sums, you don't have enough combinatorial options to hit exactly 1 without accidentally overshooting or undershooting it.

Here is the complete, exhaustive list of the 29 integers below 77 that successfully work:

• The early outliers: 11, 24

• The 30s: 30, 31, 32, 37, 38

• The 40s: 43, 45

• The 50s: 50, 52, 53, 54, 55, 57, 59

• The 60s: 60, 61, 62, 64, 65, 66, 67, 69

• The 70s: 71, 73, 74, 75, 76

(Note: The number 1 is also mathematically included in this sequence because 1/1 = 1, though it is considered a trivial case.)

The Density Pattern

When you look at the distribution of this list, you can actually see the "combinatorial tipping point" taking shape:

1. Massive initial gaps: The first non-trivial number is 11 (using 2, 3, 6). After that, the math forces you to jump 13 spots before you can successfully form 24 (using 2, 4, 6, 12).

2. Clustering begins: By the time you reach the 30s, the combinations become slightly more flexible. You start getting small, consecutive clusters of valid numbers, like 30, 31, and 32.

3. The gaps shrink: Once you cross into the 50s and 60s, the mathematical "room" available to pack in different combinations of reciprocals becomes massive. Most numbers start to work. The failures in this range (like 51, 56, 58, 63, 68, 70, 72) are just occasional skipped beats.

4. The final wall: The valid numbers pack tightly together in the mid-70s (73, 74, 75, 76). Then comes 77—the absolute final mathematical "miss"—before the overlapping combinations become so dense that they seamlessly cover every number from 78 to infinity.