An Exploration of the Arithmetic Derivative

Home , Arithmetic derivative, Cunningham chain

Alaina Sandhu Final Research Report: Summer 2006

Under Supervision of: Dr. McCallum, Ben Levitt, Cameron McLeman

1 Introduction

The arithmetic derivative is a recently defined operator on the integers whose properties directly relate to some of the most well known conjectures in number theory. While the definition of this function may have originated well into the past, perhaps the first serious analysis of the arithmetic derivative was included in the Putnam Prize Competition in 1950, and further refined by EJ Barbeau’s “Remark on an Arithmetic Derivative” [Ba61] in 1961. Upon first glance, the arithmetic derivative is a simple function defined using the unique prime factorization of integers and the product rule from calculus. This is quite deceiving, however, as the properties and behavior of the derivative are directly related to some of the oldest and most studied conjectures in elementary number theory. The arithmetic derivative operator is defined to be the unique map which sends every prime integer to 1 and that satisfies the Leibnitz rule: For all a, b ∈ Z,(ab)0 = a0b+ab0, which maintains some familiar properties from calculus such as (nk)0 = knk−1n0. Already from this definition, we can see the link to number theory. For example we can ask whether or not that for any a ∈ N, there exists a solution to the “differential equation” n0 = 2a. A proof of Goldbach’s conjecture would imply this statement, as the derivative of the product of two 0 0 primes (p1p2) is their sum (so if 2a = p1 + p2, then (p1p2) = n). As another example, we can ask whether or not there are infinitely many solutions to the differential equation n00 = 1. A proof of the Twin Prime Conjecture would imply this, since if p is a lower twin prime, then (2p)0 = p + 2 is the upper twin prime, whose derivative is thus 1 (so (2p)00 = 1). These are but a few of the related conjectures that can be explored in terms of the arithmetic derivative, and our research has uncovered yet another (see Theorem 12). There are potentially many more relationships to be explored and redefined in terms of this function, a topic requiring further research. The goal of this paper is to familiarize the reader with the properties of the arithmetic derivative and propose further conjectures with regards to the nature of the function, as well as its implications on previously established

1 conjectures. Specifically, we demonstrate conjectures which are contingent upon the existence, and characterization, of solutions to differential equations, thereby centering much of the research on the behavior and solutions of differential equations. We conclude with partial results and proposals for future work in the hope of encouraging future research of the arithmetic derivative.

2 Deﬁnitions and Background

The arithmetic derivative of a non-negative integer is deﬁned as follows: • 00 = 0. • p0 = 1 for any prime p.

• (ab)0 = a0b + ab0 for any a, b ∈ N (Leibniz rule). We now provide an explicit formula, ensuring that the function is well-deﬁned:

Qk ei Theorem 1. ([AU03], Theorem 1) For any natural number n, if n = i=1 pi is the prime factorization of n, then

k X ei n0 = n . (1) p i=1 i Qm Proof. We can write any such n by n = i=1 pi, where the pi are now no longer necessarily distinct. We proveed by induction on m. When m = 1, n is prime, and hence n0 = 1. The induction hypothesis states for any k = m ∈ N, if Qm 0 Pm 1 n = pi, then n = n . We now consider what happens when we i=1 i=1 pi add another prime pm+1:

0 0 0 (npm+1) = n pm+1 + n(pm+1) m X 1 = n + np m+1 p i=1 i m+1 X 1 = np m+1 p i=1 i

Qm ei Going back to the original expression, n = i=1 pi , our formula becomes 0 Pm ei 1 n = n , since our sum has a summand of for each power of pi dividing i=1 pi pi n, giving a total of ei . pi Lemma 1. 10 = 0. Proof. 10 = (1 · 1)0 = 10 · 1 + 1 · 10 = 2 · 10, so 10 = 0.

2 2.1 Bounds for the Arithmetic Derivative Theorem 2. [[AU03], Theorem 9] For any positive integer n

n log n n0 ≤ 2 . (2) 2

If n is composite, √ n0 ≥ 2 n. (3) Furthermore, if n is a product of k factors larger than 1, then

0 k−1 n ≥ kn k . (4)

We shall not prove this as the proof does not contain information relevant to our paper. It is important, however, to recognize that the ﬁrst derivative of every number is bounded by an explicit function of n. This will be used below in describing solutions to diﬀerential equations.

3 Diﬀerential Equations

As mentioned in the introduction, there are many conjectures within number theory that may be expressed in terms of the arithmetic derivative. As the relationships of such conjectures translates into more complex diﬀerential equations, we shall begin by solving simple equations.

3.1 Solutions to n0 = a Theorem 3. The only positive integer n which satisfies n0 = 0 is n = 1. Proof. This is a direct result of the definition of our function, since every other positive integer has at least one prime factor. Theorem 4. The only solutions to n0 = 1 in natural numbers are prime numbers. Proof. A composite number can be expressed as the product of prime numbers, of which the derivative of (by the product rule) is the sum of at least two positive integers, which is greater than 1. Now we turn our attention to the existence of solutions to the equation n0 = a, where a > 1. First, we observe from Theorem 2 that there can only be a finite number of solutions to n0 = a because all potential solutions are bounded a2 above by 4 . One direct relation of the differential equation n0 = a is the Goldbach Con- jecture, which states that every even number larger than 3 is the sum of two distinct prime numbers. In terms of our function, we can restate it as follows:

Conjecture 1. For any a ∈ N, there exists a solution to the equation n0 = 2a.

3 If the Goldbach Conjecture were proven, it would allows us to represent 0 0 0 2a = p1 +p2, so if we take the derivative of the product (p1 ·p2) = p1p2 +p1p2 = 0 p2 + p1 = 2a, and thus the differential equation n = 2a has a solution. Additionally, there exists solutions for n0 = a for certain odd numbers a: Theorem 5. If a − 2 is prime, then n0 = a has a solution, namely 2(a − 2). Proof. (2(a − 2))0 = 20(a − 2) + 2(a − 2)0 = a − 2 + 2 = a. Note, that this is not an if and only if statement; that is, there exists some numbers such that a−2 is not prime but there are still solutions to the differential equation n0 = a. If we restrict this theorem by requiring that a is an upper twin prime, the twin prime conjecture implies the following conjecture: Conjecture 2. There are infinitely many solutions to the differential equation n00 = 1. The reasoning follows directly from the above Theorem 5, since if a is an upper twin prime, its derivative will be 1.

3.2 Solutions to n0 = n There also exist unique solutions to the diﬀerential equation n0 = a, where we restrict a to be n, expressing it as n0 = n. We ﬁnd solutions to this equation are of the form, pp (where p is prime). To illustrate that these are indeed solutions:

(pp)0 = p · pp−1 · p0 = pp. (5)

This unique property of pp is seen in the following theorems: Theorem 6. ([AU03], Theorem 4) If n = pp · m for some prime p and integer 0 p 0 (k) m > 1, then n = p (m + m ) and limk→∞ n = ∞. Proof. Assume n = pp · m, then n0 = (pp)0 · m + pp · m0 = pp(m + m0) > n. Further, proof by induction shows that n(k) ≥ n + k. Theorem 7. ([AU03], Theorem 5) Let pk be the highest power of prime p that divides the natural number n. If 0 < k < p, then pk−1 is the highest power of p that divides n0. Furthermore, each derivative n, n0, n00, . . . , n(k) is distinct. Proof. Let n = pkm. Then n0 = kpk−1m+pkm0 = pk−1(km+pm0), and because k < p, the inside term is not divisible by p, therefore the entire term is only divisible by pk−1. From this argument, we can see that n00 can only be divisible by pk−2, and extending this pattern ensures that each derivative is distinct. Theorem 8. If n = ppk · m for some prime p and integers k, m > 1, then n0 = ppk(km + m0). Proof. (ppk · m)0 = pkp(pk−1) · m + ppk · m0 = ppk(km + m0)

4 Theorem 9. ([AU03], Theorem 6) For n ∈ N, n0 = n if and only if n = pp, where p is a prime. As an immediate consequence, there is an infinite number of solutions to the equation. Proof. We’ve already seen in (5), that if n = pp, n0 = pp = n. Conversely, assume n0 = n. Then by Theorem 7, if p | n at least pp | n or else it would contradict n0 = n. By Theorem 6, we conclude that this occurs when n = pp. Now that we are familiar with some differential equations and properties of our function, we shall introduce the main topic of our research and explore the associated differential equations.

4 Germain primes and Cunningham chains

A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. It is conjectured that there are infinitely many primes of this form, the largest to date being 7068555 · 2121301 − 1. A sequence of Sophie Germain primes, {p, 2p+1, 2(2p+1)+1, ...} is called a Cunningham chain of the first kind, where all but the last prime in the sequence are Sophie Germain primes. Similarly, there exists Cunningham chains of the second kind, which consist of primes of the form {p, 2p − 1, 2(2p − 1) − 1, ...}. The longest known Cunningham chains of both first and second kind were found in 2005 and are of length 16. Conjecture 3. There exists infinitely many Cunningham chains of length k, for any k ∈ N. We now explore the properties of Sophie Germain primes and Cunningham chains within the derivative. For every prime number, the following is true:

(24p)0 = 24(2p + 1). (6)

This is because (24p)0 = 4 · 23p + 24 · p0 = 24(2p + 1). Now, assuming that p is a Sophie Germain prime, we see that the derivative will yield 24(2p + 1), where now 2p + 1 is prime as well. This makes the “Sophie Germain” property detectable by diﬀerential equations: Theorem 10. For any positive integer m, we have (24m)00 ≥ 24(4m + 3), with equality if and only if m is a Sophie Germain prime.

Proof. Consider m ∈ N s.t. m is not prime. Then (24m)0 = 4 · 23m + 24 · m0 = 24(2m + m0)

(24(2m + m0))0 = 4 · 23(2m + m0) + 24(2m + m)0 = 24(4m + 2m0) + 24(2m + m0)0 = 24(4m + 2m0 + (2m + m)0) > 24(4m + 3).

5 Now consider m ∈ N where m is prime. Then (24m)0 = 4 · 23m + 24 · m0 = 24(2m + m0) = 24(2m + 1).

(24(2m + 1))0 = 25(2m + 1) + 24(2m + 1)0 = 24(4m + 2 + (2m + 1)0) ≥ 24(4m + 3), with the last inequality being an equality if and only if 2m + 1 is also prime, i.e. if and only if m is a Sophie Germain prime. So if we express n = 24p, the differential equation n00 = 4n + 48 is satisfied if and only if p is a Sophie Germain prime. Thus, a reasonable conjecture is the following: Conjecture 4. There are infinitely many solutions to the differential equation n00 = 4n + 48 where n = 24p. This conjecture is equivalent to the conjecture that there exists infinitely many Sophie Germain primes. We can now extend this pattern to Cunningham chains. Theorem 11. For any Cunningham chain of length k, (24m)(k) ≥ 24(2km + 2k − 1) with equality if and only if {m, 2m + 1, 4m + 3...} are prime. Proof. (By induction) For a Cunningham chain of length 1, (24m)0 = 4 · 23m + 24m0 = 24(2m + m0) ≥ 24(2m + 1) By Theorem 4, equality holds if and only if m is prime because m0 = 1. Assume that for a k-long Cunningham Chain, we have (24m)(k) = 24(2km + 2k − 1). Then

(24m)(k+1) = ((24)k)0 = (24(2km + 2k − 1))0 = 4 · 23(2km + 2k − 1) + 24(2km + 2k − 1)0 = 24(2k+1m + 2k+1 − 2 + 1) ≥ 24(2k+1m + 2k+1 − 1), with equality if and only if 2k+1m + 2k+1 − 1 (the k-th term in the Cunningham chain) is prime. It is further clear from this proof that this will also hold for intermediate derivatives. Making the substitution n = 24p, we conclude:

6 Theorem 12. If {p, 2p+1, ..., 2pi−1 +2i−1 −1} is a Cunningham chain of length i and n = 24p, then n(k) = 2kn + 24(2k − 1) for k = 1, 2, . . . , i. In particular, there is a bijection between Cunningham chains of length k and solutions n with n = 24p to this differential equation and These differential equations are of special significance because they allow us to characterize the existence of certain numbers and sequences in a different language. One particular re-wording is the following: Corollary 1. A Cunningham chain of length 17 exists if and only if there exists a prime number p such that n = 24p satisfies the following differential equation:

n(17) = 217n + 524272

5 Partial Results and Future Work 5.1 Bounds for the k−th derivative of all n In this section, we bound the proportion of positive integers within the interval [1, x] whose repeated derivatives tend eventually to zero. Definition 1. We define the height, ht(n), of an integer n, to be the smallest non-negative integer k such that n(k) = 0. If no such k exists, we define ht(n) = ∞. This definition allows us to phrase many interesting questions in terms of an integer’s height. For example, do there exist numbers with arbitrarily large height? We can also divide numbers into classes based on their particular height, and examine whether they share other properties other than the same height. We turn to address some of those questions now. Define the functions:

|{0 ≤ n ≤ x : ht(n) ≤ ∞}| f (x) = →0 x |{0 ≤ n ≤ x : ht(n) = k}| f (k)(x) = →0 x They are linked as follows:

0 1 k f→0(x) = f→0(x) + f→0(x) + ··· + f→0(x) + ··· , (7) where we observe that for any x, only ﬁnitely many of these terms are non-zero, so convergence is not an issue. We shall now explore this function for some values of k : We know the only number whose height is zero is zero itself, thus: 1 f (0) (x) = (8) →0 x

7 By Theorem 3, 1 is the only number whose ﬁrst derivative is zero: 1 f (1) (x) = (9) →0 x By Theorem 4, prime numbers, and only prime numbers, solve the diﬀeren- tial equation n0 = 1, and thus solve n00 = 0, so

π(x) f (2) (x) = . (10) →0 x We have seen that 2p, where p is a lower twin prime, solves the diﬀerential equation n00 = 1, so they solve n000 = 0 as well. We have an asymptotic expression for the number of twin primes ≤ x, however because we are counting the x numbers 2p, we must only consider the number of twin primes ≤ 2 :

2Π (π( x ))2 x 2 2 π2( ) x f (3) (x) > 2 ∼ 2 →0 x x 4Π (π( x ))2 = 2 2 x2 4Π ( x )2 ∼ 2 lnx x2 4Π = 2 . (ln x)2

As seen above, the expression 2p for lower twin primes will solve n000 = 0. We can integrate this expression to ﬁnd that p2, for lower twin primes, will solve n0000 = 0. This is because (p2)0 = 2p, and then we follow the same pattern above. However, this time because we are counting√ the numbers of the form p2, we can only count the number of twin primes ≤ x, to ensure we are not counting numbers exceeding x:

√ 2 √ 2Π2(π( x)) π ( x) √ f (4) (x) > 2 ∼ x →0 x x √ 2 2Π2(π( x)) = 3 x 2 √x 2Π2 (ln( x))2 ∼ 3 x 2 2Π2x = 3 1 2 2 4 x (ln(x)) 8Π = √ 2 . x(ln(x))2

To create a lower bound for f→0(x) as x → ∞, we can sum the preceding

8 expressions: √ 1 1 π(x) 4Π (π( x ))2 π ( x) f (x) > + + + 2 2 + 2 →0 x x x x2 x 2 1 4Π 8Π ∼ + + 2 + √ 2 x ln x (ln x)2 x(ln x)2 √ 2(ln x)2 + x ln x + 4Π x + 8Π x = 2 2 . x(ln x)2

We now repeat this process, bounding f→0(x) below by constructing classes of integers with infinite height, and then estimate their density: Define {|n ≤ x : ht(n) = ∞|} f (x) = = 1 − f (x). (11) →∞ x →0 The first such class of integers is the set of multiples of pp, since all of these 1 have infinite height by Theorems 6 and 9. Since in general, pp of the positive integers are divisible by pp, we can sum this quantity over all p (using inclusion exclusion to prevent overcounting) to conclude:

X |S|+1 Y p −1 f→∞(x) > (−1) (p ) S⊂P p∈S 1 1 1 1 1 1 1 = + − + − − + ··· 22 33 2233 55 2255 3355 223355 ≈ .235, for suﬃciently large x, where S ranges over all ﬁnite subsets of primes. We can further include all numbers of the form q1q2 less than or equal to x p where q1, q2 are primes that satisfy q1 + q2 ≡ 0 mod p , again for any prime p. This is because in this case:

0 p (q1q2) = q1 + q2 = p m, so ht(q1q2) = ∞, again by Theorems 6 and 9. Before we write the general formula, we will compute the term corresponding to p = 2. We are looking to ﬁnd pairs q1, q2 that satisfy q1 + q2 ≡ 0 mod 4. One way to do this is√ by choosing a prime q1 ≡ 1 mod 4 and a prime q2 ≡ 3 mod 4, where q1, q2 ≤ x (so that q1q2 ≤ x). By Dirichlet’s Theorem on arithmetic progressions, for suﬃciently large x, the primes will be evenly divided between those congruent to

1 mod 4 and 3 mod 4. Thus, asymptotically,√ the number of primes congruent p π( x) to 1 mod 4 and less than (x), is 2 , and the same follows for primes congruent to 3 mod 4.

We conclude that√ the number of pairs q1, q2 with the above properties, is π( x) 2 asymptotically ( 2 ) . For the more general formula, Dirichlet’s theorem says the primes are distributed evenly among the φ(pp) diﬀerent residue classes modulo pp. For our example, we see that half the primes are ≡ 3 mod 4 and half

9 p √ √ 2 φ(p ) π( x) 2 π( x) are ≡ 1 mod 4. This general formula then becomes 2 ( φ(pp) ) = 2φ(pp) . Summing over all p, and again using inclusion-exclusion, we conclude (along with the above result) that √ X Y 1 π( x)2 X Y 1 f (x) > (−1)|S|+1 + (−1)|S|+1 →∞ pp 2 φ(pp) S⊂ p∈S S⊂ p∈S P √ P ≈ .235 + .269π( x)2, for suﬃciently large x. Better estimates for both bounds can be obtained through further research into the patterns and behavior of the arithmetic derivative.

5.2 Derivative of a primorial Another interesting result is the implication for derivatives of primorial numbers. The n-th primorial is defined to be the product of all primes up through n. Qk Definition 2. Let n = i=1 pi for every prime number, so n represents the 0 Qk Pk 1 k−th primorial. Then n = pi . i=1 i=1 pi One interesting relation to this definition is its relation to the infinitude of primes: Consider, for example, the k-th primorial n. Because n contains each consecutive prime up through pn raised only to the first power, by Theorem 7, none of the preceding primes can divide n0. We shall prove this as follows:

Proof. Suppose {p1, p2, . . . , pk} are the only primes, and we represent n = Qk 0 i=1 pi. When taking the derivative n , Theorem 7 restricts all the primes {p1, p2, . . . , pk} from dividing the derivative, because they are all raised to the 0 first power. So n has at least one prime factor not included in {p1, p2, . . . , pk}, leading us to a contradiction. Therefore, we conclude that there are infinite primes. It would be interesting to explore this property to find implications it may have on the behavior of very large primes. Throughout our paper we have show, and conjectured, that there exists many other primes, and sequences of primes, whose properties can be translated to the language of differential equations and arithmetic derivatives. We have already encountered one prime sequence, namely Cunningham chains, which could be defined and explored within our function. Also, we believe there may be other properties in our derivative that lie in the pattern of prime arithmetic progressions.

10 6 Appendix

The following are codes for various functions with the arithmetic derivative. The comment line above each code describes its function. This code will compute the arithmetic derivative of any integer or rational number:

{f(n)=sign(n)*abs(n)*sum(i=1, matsize(factor(abs(n)))[1],}

{factor(abs(n))[i,2]/factor(abs(n))[i,1])}

This code will compute the solutions, a for a0 = n:

{I(n) = for(i=1, n^2/4+1, if((f(i))==n, print(i)));} This code computes the number of solutions, a, to solve a0 = n:

{ i(n)= count = 0; k=n; for(i=1, k^2/4+1, if((f(i))== k, count = count + 1)); print(count);}

11 7 Table of n(k) for n ≤ 100 and k ≤ 10

n n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) n(9) n(10) 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 5 1 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 8 12 16 32 80 176 368 752 1520 3424 8592 9 6 5 1 0 0 0 0 0 0 0 10 7 1 0 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 0 12 16 32 80 176 368 752 1520 3424 8592 20096 13 1 0 0 0 0 0 0 0 0 0 14 9 6 5 1 0 0 0 0 0 0 15 8 12 16 32 80 176 368 752 1520 3424 16 32 80 176 752 1520 3424 8592 20096 70464 235072 17 1 0 0 0 0 0 0 0 0 0 18 21 10 7 1 0 0 0 0 0 0 19 1 0 0 0 0 0 0 0 0 0 20 24 44 48 112 240 608 1552 3120 8144 16304 21 10 7 1 0 0 0 0 0 0 0 22 13 1 0 0 0 0 0 0 0 0 23 1 0 0 0 0 0 0 0 0 0 24 44 48 112 240 608 1552 3120 8144 16304 32624 25 10 7 1 0 0 0 0 0 0 0 26 15 8 12 16 32 80 176 368 752 1520 27 27 27 27 27 27 27 27 27 27 27 28 32 80 176 368 752 1520 3424 27 27 70464 29 1 0 0 0 0 0 0 0 0 0 30 31 1 0 0 0 0 0 0 0 0 31 1 0 0 0 0 0 0 0 0 0 32 80 176 368 752 1520 3424 8592 20096 70464 235072 33 14 9 6 5 1 0 0 0 0 0 34 19 1 0 0 0 0 0 0 0 0 35 12 16 32 80 176 368 752 1520 3424 8592 36 60 92 96 272 560 1312 3312 8976 22288 47872 37 1 0 0 0 0 0 0 0 0 0 38 21 10 7 1 0 0 0 0 0 0 39 16 32 80 176 368 752 1520 3424 8592 20096 40 68 72 156 220 284 288 912 2176 7744 24640 41 1 0 0 0 0 0 0 0 0 0 42 41 1 0 0 0 0 0 0 0 0 43 1 0 0 0 0 0 0 0 0 0 44 48 112 240 608 1552 3120 8144 16304 32624 65264 45 39 16 32 80 17612 368 752 1520 3424 8592 n n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) n(9) n(10) 46 25 10 7 1 0 0 0 0 0 0 47 1 0 0 0 0 0 0 0 0 0 48 112 240 608 1552 3120 8144 16304 32624 65264 130544 49 14 9 6 5 1 0 0 0 0 0 50 45 39 16 32 80 176 368 752 1520 3424 51 20 24 44 48 112 240 608 1552 3120 8144 52 56 92 96 272 560 1312 3312 8976 22288 47872 53 1 0 0 0 0 0 0 0 0 0 54 81 108 216 540 1188 2484 5076 10260 23112 57996 55 16 32 80 176 368 752 1520 3424 8592 20096 56 92 96 272 560 1312 3312 8976 22288 47872 198656 57 22 13 1 0 0 0 0 0 0 0 58 31 1 0 0 0 0 0 0 0 0 59 1 0 0 0 0 0 0 0 0 0 60 92 96 272 560 1312 3312 8976 22288 47872 198656 61 1 0 0 0 0 0 0 0 0 0 62 33 14 9 6 5 1 0 0 0 0 63 51 20 24 44 48 112 240 608 1552 3120 64 192 640 2368 7168 36864 245760 1851392 12976128 120127488 1012858880 65 18 21 10 7 1 0 0 0 0 0 66 61 1 0 0 0 0 0 0 0 0 67 1 0 0 0 0 0 0 0 0 0 68 72 156 220 284 288 912 2176 7744 24640 84608 69 26 15 8 12 16 32 80 176 368 752 70 59 1 0 0 0 0 0 0 0 0 71 1 0 0 0 0 0 0 0 0 0 72 156 220 284 288 912 2176 7744 24640 84608 296256 73 1 0 0 0 0 0 0 0 0 0 74 39 16 32 80 176 368 752 1520 3424 8592 75 55 16 32 80 176 368 752 1520 3424 8592 76 80 176 368 752 1520 3424 8592 20096 70464 235072 77 18 21 10 7 1 0 0 0 0 0 78 71 1 0 0 0 0 0 0 0 0 79 1 0 0 0 0 0 0 0 0 0 80 176 368 752 1520 3424 8592 20096 70464 235072 705280 81 108 216 540 1188 2484 5076 10260 23112 57996 135648 82 43 1 0 0 0 0 0 0 0 0 83 1 0 0 0 0 0 0 0 0 0 84 124 128 448 1408 5056 15232 56384 169216 677120 2902784 85 22 13 1 0 0 0 0 0 0 0 86 45 39 16 32 80 176 368 752 1520 3424 87 32 80 176 368 752 1520 3424 8592 20096 70464 88 140 188 192 640 2368 7168 36864 245760 1851392 12976128 89 1 0 0 0 0 0 0 0 0 0 90 123 44 48 ‘ 240 608 1552 3120 8144 16304 91 20 24 44 48 112 240 608 1552 3120 8144 92 96 272 560 1312 3312 8976 22288 47872 198656 1094656 13 93 34 19 1 0 0 0 0 0 0 0 94 49 14 9 6 5 1 0 0 0 0 95 24 44 48 112 240 608 1552 3120 8144 16304 n n(1) n(2) n(3) n(4) n(5) n(6) n(7) n(8) n(9) n(10) 96 272 560 1312 3312 8976 22288 47872 198656 1094656 5474304 97 1 0 0 0 0 0 0 0 0 0 98 77 18 21 10 7 1 0 0 0 0 99 75 55 16 32 80 176 368 752 1520 3424 100 140 188 192 640 2368 7168 36864 245760 1851392 12976128

References

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