The Personality of the Integers from One to One Gross

Home , Abundant number, Deficient number, Friendly number

ETY CI O O F S The Dozenal Society of America E 0 1 A L X 2 M

9 3 E The Personality of the Integers

8 4 I

7 5 C from One to One Gross A

O 6 D Jay Schiffman S

Introduction

n his fascinating paperback entitled The Lore of integer that lends a special personality to them es- I Large Numbers,1 author Philip J. Davis devotes a pecially when one resorts to duodecimals. Finally, detailed chapter to The Personality of Numbers and my goal is to furnish number theoretic characteris- provides a list entitled Who’s Who Among The Inte- tics related to each of the integers that are perhaps gers from One to One Hundred in the decimal base. omitted from the average list. Our initial task is to The purpose of my paper is to expand his list to the present the reader with a dictionary which is termed first gross and present additional characteristics of A Glossary of Symbols and Terms. These symbols these integers. It is indeed the uniqueness of each and terms will be illustrated in this section.

A Glossary of Symbols and Terms

1. n! The factorial of the positive integer n. We de- 6. Ln The nth Lucas Number. n is a Lucas Number fined n!, known as n factorial, recursively as if n satisfied the Lucas Sequence defined recur- follows: 0! = 1 and (n + 1)! = (n + 1)(n!). sively as follows: L1 = 1, L2 = 3, and Ln = Hence 1! = (0 + 1)(0!) = (1)(1) = 1 and Ln−2 + Ln−1 for n ≥ 3. Hence L3 = L1 + L2 5! = (4 + 1)(4!) = (5)(20) = X0. = 1 + 3 = 4.

2. Tn The nth Triangular Number. We defined 7. pn The nth Prime Number. Define a prime num- the nth Triangular Number Tn as follows: Tn ber to be any integer p > 1 that has only 1 and = 1 + 2 + 3 + ... + n = ((n)(n + 1)/2. Thus T5 p as its factors (divisors). Hence p3 = 5; for the = 1 + 2 + 3 + 4 + 5 = 13 = ((5)(5 + 1))/2. only divisors of 5 are 1 and 5.

8. FEn The nth Fermat Prime. n is a Fermat Prime 3. Sn The nth Square Number. Define the nth n if n is prime and n = 22 +1 for any whole num- Square Number Sn by the formula Sn = 1 + 2 2 2 3 + 5 + 7 + ... + (2n − 1) = n . Hence S4 ber n. To cite an example, FE2 = 2 + 1 = 15. = 1 + 3 + 5 + 7 = 14 = 42. 9. COn The nth Integer for which a regular n-gon

4. THn The nth Tetrahedral Number. The is constructible using solely a straight-edge and Tetrahedral Numbers form the sequence compass. This idea is related to the geometric 1, 4, X, 18, 2E, 48,.... The Tetrahedral Numbers notion of Constructible Regular n-gons in the are related to the Triangular Numbers as fol- following sense as can be demonstrated via ab- n lows: THn = THn−1 + Tn for n ≥ 2 and TH1 stract algebraic techniques: The sole values of n = 1. Hence TH5 = TH4 + T5 = 18 + 13 = 2E. for which a regular -gon is constructible using only a straight-edge and compass are such that 5. Fn The nth Fibonacci Number. n is a Fibonacci the only odd primes dividing n are the Fermat Number if n satisfies the Fibonacci Sequence Primes whose squares do not divide n. Hence defined recursively as follows: F1 = 1, F2 = 1, a regular duodecahedron [sic] (10 sides) is con- 2 and Fn = Fn−2 + Fn−1 for n ≥ 3. Hence F3 = structible in the above sense; for 10 = (2 )(3) F1 + F2 = 1 + 1 = 2. and 3 is a Fermat Prime (F1 = 3 as the reader 1Philip J. Davis, The Lore of Large Numbers. The New Mathematical Library, New York, 1961.

1 can readily verify) but 32 is not a factor of 10. that the sum of all the proper divisors of n is On the other hand, a regular 16-gon is not con- greater than n if n is abundant. It can be shown structible since 16 = (2)(32) and while 3 is a that any integer multiple of a perfect number is Fermat Prime, 32 is a factor of 16. necessarily abundant (10 fits this description) while most abundant numbers are even. In fact, X. Cn The nth Composite Number. n is designated the initial odd abundant number is 669 (deci- Composite if n > 1 and n is not prime. To cite mally nine hundred forty-five), and we observe an example, 10 is composite; for 10 possesses a that σ(669) = 1140 > (2)(669) = 1116. In fact, half dozen factors: 1, 2, 3, 4, 6, and 10 while 1 + 3 + 5 + 7 + 9 + 13 + 19 + 23 + 2E + 39 + 53 + prime numbers possess only a pair of factors (1 89 + E3 + 139 + 223 = 693 > 669. and the prime itself). 14. Dn The nth Deficient Number. A positive inte- E. HCn The nth Hypercomposite Number. A pos- ger n is classified as Deficient if it is less than itive integer n is termed Hypercomposite if n the sum of its aliquot divisors. To illustrate, 2E possesses a greater number of factors than any is deficient, since 1+5+7 = 11 < 2E. The equa- of its immediate predecessors. Again 10 serves tion σ(n) < 2n signifies that n is deficient; that as a near example; for 10 has 6 factors while no is, the sum of all the divisors of n including n is integer < 10 has more than 4 factors (6, 8, and smaller than twice n. Equivalently, one gener- X). ally writes the inequality σ0(n) < n to indicate 10. SFn The nth Square-Free Number. n is Square- that the sum of the proper divisors of n is less Free if n is not divisible by the square of any than n if n is deficient. It can be shown with prime p. To illustrate, 26 = (2)(3)(5) en- relative ease that any prime number is deficient. joys this property while our favorite integer Moreover, so is any power of a prime and any 10 = (22)(3) does not having 22 as a part of integer save for 6 = (2)(3) that is expressible as its factorization. the product of two distinct primes.

11. σ(n) The sum of the divisors of a positive in- 15. Mn The nth Mersenne Prime. n is a Mersenne teger n including n itself. To cite some ex- Prime is n is prime and n = 2p − 1, where p ample, σ(6) = 1 + 2 + 3 + 6 = 10, while itself is prime. Mersenne primes lead to per- (σ(10) = 1 + 2 + 3 + 4 + 6 + 10 = 24) and fect numbers. In fact, the form of an even σ(11) = 1 + 11 = 12. perfect number according to the Greek Math- ematician Euclid is p = (2p−1)(2p − 1) where P n 12. n The th Perfect Number. A positive integer both p and 2p−1 are prime. To illustrate, if n is styled Perfect if it is equal to the sum of all we take p = 3, then 3 is prime and so is its aliquot (proper) divisors. For example, 6 s 23 − 1 = 7. This leads to the second perfect 6 = 1+2+3 perfect since . In short, any integer number P = (23−1)(23 − 1) = (22)(7) = 24. which coincides with the sum of all its divisors not including itself is classified as perfect. Per- 16. τ(n) The number of divisors of a positive integer fect numbers are rare indeed. The next three n. To cite some examles, τ(10) = 6 (1, 2, 3, 4, perfect numbers in order of magnitude are 24, 6, 10 are the divisors of 10) while τ(6) = 4 (1, 354, and 4854. Mathematicians often succinctly 2, 3, and 6 are the divisors of 6) and τ(5) = 2 (1 employ the equation σ(n) = 2n to denote that and 5 are the divisors of 5). It is readily appar- n is perfect; that is, the sum of all the divi- ent that if p is a prime, then τ(p) = 2, while for sors of n including n itself is precisely twice n. any power of a prime p, say p2, τ(p2) = n + 1. 3 Equivalently, one could write σ0(n) = n to con- For example, 8 = 2 and τ(8) = 4; for 1, 2, 4, note that the sum of all the proper divisors of and 8 are the divisors of 8. n is precisely n if n is perfect. 17. φ(n) The number of positive integers < n that 13. An The nth Abundant Number. A positive in- are relatively prime to n. To illustrate, φ(10) = teger n is termed Abundant if it is greater than 4 since 1, 5 7, and E are < 10 and have no fac- the sum of all its aliquot divisors. For example, tors in common with 10 other than 1, which is 10 is abundant, since 1+2+3+4+6 = 14 > 10. equivalent to saying that the integers are rela- The equation σ(n) > 2n indicates that n is tively prime to 10. In addition, φ(7) = 6; for abundant; that is, the sum of all the divisors 1, 2, 3, 4, 5, and 6 are all < 7 and relatively of n including n is larger than twice n. Equiv- prime to 7. It is readily apparent that if p is 2 alently, one often writes σ0(n) > n to signify prime, then φ(p) = p − 1. Moreover, if t = p

2 (p a prime), then φ(pn) = pn − pn−1. Hence of a multiply-perfect number. We define n to φ(8) = φ(23) = 23 −23−1 = 23 −22 = 8−4 = 4. be multiply-perfect if σ(n) = (k)(n) for some Observe that this agrees with enumeration; for positive integer k ≥ 3. (A perfect number is in the integers 1, 3, 5, and 7 are < 8 and are essence a multiply perfect number with k = 2.) relatively prime to 8. This number-theoretic Our initial example of a 3-perfect number is X0. function known as the Euler-phi function or σ(X0) = 1 + 2 + 3 + 4 + 5 + 6 + 8 + X + 10 + 13 + totient function possesses extremely important 18+20+26+34+50+X0 = 260 = (3)(X0). The applications in higher algebra and the theory of reader is invited to show that 480 is a second numbers. In fact, recalling our discussion ear- 3-perfect prime number; 4-perfect numbers are lier concerning constructible regular polygons also known although the smallest one is quite utilizing only a straight-edge and compass, one large. can demonstrate that the values of n for which a regular n-gon is constructible using only a 20. A(m, n) An amicable or friendly number pair. straight-edge and compass are those values for A pair of numbers m and n are termed ami- which [φ(n)/2] (and thus φ(n)) is an integral cable or friendly if the sum of the aliquot di- power of 2. Hence a regular duodecagon is con- visors of one is equal to the other. In essence, structible since φ(10) = 4 = 22 as is a regular we are asserting that A(m, n) is an amicable 2 octagon (φ(8) = 4 = 2 ) while a regular 16-gon pair if σ0(m) = n and σ0(n) = m. Moreover is not (φ(16) = 6 and 6 is not an integral power from this, one deduces that if A(m, n) is an of 2). amicable pair, then σ(m) = σ(n) = m + n. If m = 164 and n = 1E8, then σ0(164) = 18. π(n) The number of positive primes ≤ n. To cite 1+2+4+5+X+E+18+1X+38+47+92 = 1E8 an example, observe hat π(10) = 5, since 2, 3, σ (1E8) = 1 + 2 + 4 + 5E + EX = 164 E while 0 , so 5, 7, and are primes not exceeding a dozen. that (164, 1E8) is an amicable number pair, the π(11) = 6 E Morever, ; for 2, 3, 5, 7, , and 11 are smallest of its kind. Note that σ(164) = 360 = ≤ 11 primes . σ(1E8) = 164 + 1E8. The reader is invited to 19. Πd|n The product of the divisors of a pos- check that the number pair (828, 84X) is a sec- itive integer n. To illustrate, Πd|10 = ond amicable number pair, discovered in 10E6, (1)(2)(3)(4)(6)(10) = 1000 = 103, while by a 14; year old Italian school boy. The pair Πd|23 = (1)(3)(9)(23) = 509 and Πd|2E = seemingly escaped the great mathematicians of (1)(5)(7)(2E) = 861. Observe in the latter pair their day. of examples that 23 and 2E coincide with the product of their proper divisors. It should be noted in closing that σ(n), τ(n), and φ(n) are multiplicative number-theoretic functions in 1X. PSKn The nth Perfect Number of the Second the sense that for relatively prime pairs of integers Kind. The idea of a Perfect Number of the r and s, σ(rs) = σ(r) × σ(s), τ(rs) = τ(r) × τ(s), Second Kind generalizes the notion of a per- and φ(rs) = φ(r) × φ(s). This result is extendable fect number. n is styled Perfect of the Second to more than two integers provided none of the inte- Kind if the product of the divisors of n coin- gers has a common factor apart from 1 among them. cides with the square of the number n. Symbol- To cite an example, we calculate σ(X0), τ(X0), and ically, one would express this fact via the equa- φ(X0). Initially observe that X0 = (23)(3)(5). Now Πd|n = n2 E tion . Citing the examples 23 and 2 σ(X0) = σ[(23)(3)(5)] = (23) × σ(3) × σ(5) = [(23+1 − from 19 above illustrates this type of behavior; 1)/(2 − 1)] × (3 + 1) × (5 + 1) = [(24 − 1)/(2 − 1) = 509 = 232 861 = 2E2 for and . Moreover, it is (14 − 1)/(2 − 1) × (4) × (6) = (13)/(1) × (4) × (6) = immediate that if n = (p)(q) for distinct primes (13) × (4) × (6) = 260, τ(X0) = [(23)(3)(5)] = τ(23) × p and q or if n = p3 for some prime p, then the τ(3)×τ(5) = (3+1)×(1+1)×(5+1) = 4×2×2 = 14, product of all the aliquot divisors of n coincides and φ(X0) = [(23)(3)(5)] = [23 − 23−1] × (3 − 1) × (5 − n n with and the Product of all the divisors of is 1) = [23 − 22] × 2 × 4 = (8 − 4) × (3 − 1) × (5 − 1) = n2. (Note that (1)(p)(q)(pq) = (p2)(q2) = (pq)2 4 × 2 × 4 = 28. and (1)(p)(p2)(p3) = (p6) = (p3)2.) Our next goal is to provide an enumeration of a 1E. MPn The nth Multiply Perfect Number. A sec- number of essential dozenal sets of numbers in the ond generalization of a perfect number is that range from 1 to 100.

3 Dozenal Sets of Numbers (Range 1–100)

1. Perfect Numbers {6, 24}. (Total: 2) 9. Fibonacci Numbers {1, 1, 2, 3, 5, 8, 11, 19, 2X, 47, 75, 100}. (Total: 10) 2. Deficient Numbers {1, 2, 3, 4, 5, 7, 8, 9, X, E, 11, 12, 13, 14, 15, 17, 19, 1X, 1E, 21, 22, 23, 25, X. Lucas Numbers {1, 3, 4, 7, E, 16, 25, 3E, 64, 27, 28, 29, 2X, 2E, 31, 32, 33, 35, 37, 38, 39, 3X, X3}. (Total: X) 3E, 41, 42, 43, 44, 45, 47, 49, 4X, 4E, 51, 52, 53, 54, 55, 57, 58, 59, 5X, 61, 62, 63, 64, 65, 67, 69, E. Composite Numbers {4, 6, 8, 9, X, 10, 12, 13, 6X, 6E, 71, 72, 73, 75, 77, 78, 79, 7X, 7E, 81, 82, 14, 16, 18, 19, 1X, 20, 21, 22, 23, 24, 26, 28, 29, 83, 85, 87, 89, 8X, 8E, 91, 92, 93, 95, 97, 98, 99, 2X, 2E, 30, 32, 33, 34, 36, 38, 39, 3X, 40, 41, 42, 9X, 9E, X1, X2, X3, X4, X5, X7, X8, X9, XX, XE, E1, 43, 44, 46, 47, 48, 49, 4X, 50, 52, 53, 54, 55, 56, E2, E3, E4, E5, E7, E9, EX, EE}. (Total: 91) 58, 59, 5X, 60, 62, 63, 64, 65, 66, 68, 69, 6X, 70, 71, 72, 73, 74, 76, 77, 78, 79, 7X, 7E, 80, 82, 83, 3. Abundant Numbers {10, 16, 18, 20, 26, 30, 34, 84, 86, 88, 89, 8X, 90, 92, 93, 94, 96, 97, 98, 99, X 36, 40, 46, 48, 50, 56, 5 , 60, 66, 68, 70, 74, 76, 9X, 9E, X0, X1, X2, X3, X4, X5, X8, X9, XX, E0, E1, X X E E E 80, 84, 86, 88, 90, 94, 96, 0, 6, 0, 6, 8, E2, E3, E4, E6, E8, E9, EX, EE, 100}. (Total: 91) 100} (Total: 29) 10. Perfect Numbers of the Second Kind {6, 4. Multiply Perfect Numbers {X0} (3-Perfect 8, X, 12, 13, 19, 1X, 22, 23, 29, 2X, 2E, 32, 33, Number) (Total: 1) 3X, 43, 47, 49, 4X, 52, 55, 59, 62, 65, 6X, 71, 72, 5. Square-Free Numbers {1, 2, 3, 5, 6, 7, X, E, 11, 73, 77, 79, 7X, 7E, 8X, 93, 97, 9X, 9E, X2, X3, X5, 12, 13, 15, 17, 19, 1X, 1E, 22, 25, 26, 27, 29, 2X, X9, E1, E2, E9, EX, EE}. (Total: 3X) 2E, 31, 32, 33, 35, 36, 37, eX, 3E, 43, 45, 47, 49, 4X, 4E, 51, 52, 55, 56, 57, 59, 5X, 5E, 61, 62, 65, 11. Set of Values such that a Regular n-gon 66, 67, 6X, 6E, 71, 72, 73, 75, 77, 79, 7X, 7E, 81, is constructible using only a straight-edge and X 85, 86, 87, 89, 8X, 8E, 91, 92, 93, 95, 96, 97, 9X, compass: {3, 4, 5, 6, 8, , 10, 13, 14, 15, 18, 20, X 9E, X2, X3, X7, X9, XX, XE, E1, E2, E5, E6, E7, E9, 26, 28, 2 , 34, 40, 43, 50, 54, 58, 68, 71, 80, 86, X X E EX, EE}. (Total: 75) 0, 8, 4}. (Total: 24) 6. Prime Number {2, 3, 5, 7, E, 11, 15, 17, 1E, 25, 12. Fermat Primes {3, 5, 15}. (Total: 3) 27, 31, 35, 37, 3E, 45, 4E, 51, 57, 5E, 61, 67, 6E, 75, 81, 85, 87, 8E, 91, 95, X7, XE, E5, E7}. 13. Hypercomposite Numbers {2, 4, 8, 10, 20, X (Total: 2X) 30, 40, 50, 0}. (Total: 9)

7. Triangular Numbers {1, 3, 6, X, 13, 19, 24, 30, 14. Mersenne Primes {M2 = 3, M3 = 7, M5 = X 39, 47, 56, 66, 77, 89, X0, E4} (Total: 14) 27, M7 = 7}. (Total: 4) 8. Tetrahedral Numbers {1, 4, X, 18, 2E, 48, 70, 15. Square Numbers {1, 4, 9, 14, 21, 30, 41, 54, X0}. (Total: 8) 69, 84, X1, 100}. (Total: 10)

The Catalogue of Integers from One to One Gross Characteristics

n what follows, we shall make use of the sym- 1. Unit; Multiplicative Identity; Neither Prime nor F n I bols described in our glossary above. Hence n Composite. F1; F2; 1 for all counting integers will denote the nth Fibonacci Number and Cn will n; 0!; 1!; T1; TH1; L1; S1; SF1; D1; τ(1) = 1; connote the nth Composite Number. Thus F5 = 5, φ(1) = 1; σ(1) = 1; π(1) = 1; Πd|1 = 1. since the fifth Fibonacci Number in the sequence is five, while CX = 16, as the tenth composite integer in n n the sequence is one dozen six. The other codes are 2. p1; Binary Base; 1 + 1 for all counting integers obtained in similar fashion, and we cap the article n; HC1; F3; 2!; F1 + F2; D2; SF2; τ(2) = 2; with the catalogue. φ(2) = 1; σ(2) = 3; π(2) = 1; Πd|2 = 2.

4 3. p2; Ternary Base; M2; FE1; C1; F4; T2; L2; 1! 17. p8; D14; SF11; τ(17) = 2; φ(17) = 16; σ(17) = + 2!; F1 + F3; SF3; D3; τ(3) = 2; φ(3) = 2; 18; π(17) = 8; Πd|17 = 17. σ(3) = 4; π(3) = 2; Πd|3 = 3. 2 2 2 18. 2 × 5; A3; CE; score; TH4; 2 + 4 ; F1 + F2 2 4. 2 ; C1; HC2; TH2; L3; S2; CO2; F1 + F2 + F3; + F3 + F4 + F5 + F6; τ(18) = 6; φ(18) = 8; F2 + F4; D4; L1 + L2; τ(4) = 3; φ(4) = 2; σ(18) = 36; π(18) = 8; Πd|18 = 4768. σ(4) = 7; π(4) = 2; Πd|4 = 8. 19. 3 × 7; C10; PSK6; F8; SF12; T6; F1 + F3 + F5 2 2 + F7; D15; τ(19) = 4; φ(19) = 10; σ(19) = 28; 5. p3; 2 + 1 ; FE2; F5; CO3; SF4; D5; L1 + L3; π(19) = 8 Πd|19 = 309 τ(5) = 2; φ(5) = 4; σ(5) = 6; π(5) = 3; ; . Πd|5 = 5 . 1X. 2 × E; C11; PSK7; SF13; D16; τ(1X) = 4; φ(1X) = X; σ(1X) = 30; π(1X) = 8; Πd|1X = 344. 6. 2 × 3; P1; C2; PSK1; SF5; CO4; T3; 3!; τ(6) = 4; φ(6) = 2; σ(6) = 10; π(6) = 3; Πd|6 = 30. 1E. p9; SF14; D17; τ(1E) = 2; φ(1E) = 1X; σ(1E) = 20; π(1E) = 9; Πd|1E = 1E. 7. p4; L4; F1 + F2 + F3 + F4; M3; SF6; D6; τ(7) = 2; 3 φ(7) = 6; σ(7) = 8; π(7) = 4; Πd|7 = 7. 20. 2 × 3; HC5; C12; A4; CO10; 4!; τ(20) = 8; φ(20) = 8; σ(20) = 50; π(20) = 9; Πd|20 = 3 2 2 8. 2 ; C3; PSK2; Octal Base; F6; 2! + 3!; 2 + 2 ; 14 0000. CO5; F1 + F3 + F5; D7; L1 + L2 + L3; HC3; 2 2 2 τ(8) = 4; φ(8) = 4; σ(8) = 13; π(8) = 4; 21. 5 ; 3 + 4 ; C13; D18; S5; τ(21) = 3; φ(21) = 18; X Πd|8 = 54. σ(21) = 27; π(21) = 9; Πd|21 = 5. 2 2 2 3 3 22. 2 × 11; SF15; C14; 1 + 5 ; PSK8; D19; L1 + 9. 3 ; C4; S3; 1 + 2 ; 1! + 2! + 3!; D8; τ(9) = 3; L2 + L3 + L4 + L5; τ(22) = 4; φ(22) = 10; φ(9) = 6; σ(9) = 11; π(9) = 4; Πd|9 = 23. σ(22) = 36; π(22) = 9; Πd|22 = 484. 2 2 X. 2×5; C5; CO6; PSK3; SF7; Decimal Base; 1 +3 ; 3 23. 3 ; PSK9; C15; D1X; τ(23) = 4; φ(23) = 16; T TH D L L τ(X) = 4 φ(X) = 4 4; 3; 9; 2 + 4; ; ; σ(23) = 34; π(23) = 9; Πd|23 = 509. σ(X) = 16; π(X) = 4; Πd|X = 84. 2 3 3 24. 2 × 7; P2; C16; 1 + 3 ; T7; L2 + L4 + L6; E E E X E . p5; SF8; L5; DX; τ( ) = 2; φ( ) = ; σ( ) = 10; τ(24) = 6; φ(24) = 10; σ(24) = 48; π(24) = 9; π(E) = 5; Πd|E = E. Πd|24 = 1 0854. 2 2 2 2 2 2 10. 2 × 3; Duodecimal Base; HC4; C6; CO7; A1; 2! 25. pX; 2 +5 ; L7; SF16; D1E; 2 +3 +4 ; τ(25) = 2; × 3!; F1 + F2 + F3 + F4 + F5; F2 + F4 + F6; φ(25) = 24; σ(25) = 26; π(25) = X; Πd|25 = 25. τ(10) = 6; φ(10) = 4; σ(10) = 28; π(10) = 5; 2 × 3 × 5 C A CO 12 + 22 + 32 + 42 SF Πd|10 = 1000. 26. ; 17; 5; 11; ; 17; 3! + 4!; τ(26) = 8; φ(26) = 8; σ(26) = 60; 2 2 X 11. p6; F7; 2 + 3 ; DE; SF9; τ(11) = 2; φ(11) = 10; π(26) = ; Πd|26 = 33 0900. σ(11) = 12; π(11) = 6; Πd|11 = 11. 27. pE; M5; SF18, D20; τ(27) = 2; φ(27) = 26; 2 2 2 σ(27) = 28; π(27) = E; Πd|27 = 27. 12. 2×7; C7; SFX; PSK4; 1 +2 +3 ; D10; τ(12) = 4; φ(12) = 6 σ(12) = 20 π(12) = 6 Πd|12 = 144 5 4 4 2 2 ; ; ; . 28. 2 ; C18; 2 +2 ; 4 +4 ; CO12; D21; 2! + 3! + 4!; τ(28) = 6; φ(28) = 14; σ(28) = 53; π(28) = E; 13. 3 × 5; C8; PSK5; CO8; T5; D11; L1 + L2 + Πd|28 = 16E68. L3 + L4; τ(13) = 4; φ(13) = 8; σ(13) = 20; 5 5 π(13) = 6; Πd|13 = 179. 29. 3 × E; SF19; C19; PSKX; D22; 1 + 2 ; 1! + 2! + 3! 4! F F F F F F F 4 2 3 3 + ; 2 + 4 + 6 + 8; 1 + 2 + 3 14. 2 ; Hexadecimal Base; 4 ; S4; C9; 2 + 2 ; CO9; + F4 + F5 + F6 + F7; τ(29) = 4; φ(29) = 18; D L L L τ(14) = 5 φ(14) = 8 12; 1 + 3 + 5; ; ; σ(29) = 40; π(29) = E; Πd|29 = 769. σ(14) = 27; π(14) = 6; Πd|14 = 714. 2 2 2X. 2 × 15; SF1X; D23; C1X; PSKE; 3 + 5 ; F9; CO13; 2 2 4 4 15. p7; 1 + 4 ; 1 + 2 ; FE3; COX; D13; SF10; τ(2X) = 4; φ(2X) = 14; σ(2X) = 46; π(2X) = E; τ(15) = 2; φ(15) = 14; σ(15) = 16; π(15) = 7; Πd|2X = 804. Πd|15 = 15. 2 2 2 3 3 2E. 5×7; SF1E; D24; C1E; PSK10; 1 +3 +5 ; 2 +3 ; 2 2 2 16. 2 × 3 ; CX; A2; 3 + 3 ; L6; τ(16) = 6; φ(16) = TH5; τ(2E) = 4; φ(2E) = 20; σ(2E) = 40; 6;σ(16) = 33; π(16) = 7; Πd|16 = 3460. π(2E) = E; Πd|2E = 861.

5 2 2 3 3 3 2 2 2 30. 2 × 3 ; C20; HC6; 1 + 2 + 3 ; 6 ; S6; T8; A6; 45. p14; SF29; D36; 2 + 7 ; τ(45) = 2; φ(45) = 44; τ(30) = 9; φ(30) = 10; σ(30) = 77; π(30) = E; σ(45) = 46; π(45) = 14; Πd|45 = 45. Πd|30 = 346 0000 . 3 3 3 2 2 2 2 46. 2 × 3 ; 3 + 3 ; C31; AX; 2 + 3 + 4 + 5 ; 2 2 31. p10; SF20; D25; 1 + 6 ; τ(31) = 2; φ(31) = 30; F1 + F2 + F3 + F4 + F5 + F6 + F7 + F8; σ(31) = 32; π(31) = 10; Πd|31 = 31. τ(46) = 8; φ(46) = 16; σ(46) = X0; π(45) = 14; Πd|46 = 2X2 0900. 32. 2 × 17; SF21; C21; D26; PSK11; τ(32) = 4; 2 2 φ(32) = 16; σ(32) = 50; π(32) = 10; Πd|32 = 47. 5 × E; C32; SF2X; D37; PSK15; FX; 1 + 2 + 2 2 2 X04. 3 + 4 + 5 ; TX; F1 + F3 + F5 + F7 + F9; τ(47) = 4; φ(47) = 34; σ(47) = 60; π(47) = 14; 33. 3 × 11; SF22; C22; D27; PSK12; τ(33) = 4; Πd|47 = 1901. φ(33) = 20; σ(33) = 48; π(33) = 10; Πd|33 = 3 2 2 2 X69. 48. 2 × 7; C33; AE; 2 + 4 + 6 ; TH6; τ(48) = 8; φ(48) = 20; σ(48) = X0; π(48) = 14; Πd|48 = 3 2 2 34. 2 ×5; C23; CO14; 2 +6 ; A7; τ(34) = 8; φ(34) = 336 3314. 14; σ(34) = 76; π(34) = 10; Πd|34 = X3 5594. 49. 3 × 17; C34; SF2E; PSK16; D38; τ(49) = 4; 2 2 35. p11; SF23; D23; 4 + 5 ; τ(35) = 2; φ(35) = 34; φ(49) = 30; σ(49) = 68; π(49) = 14; Πd|49 = σ(35) = 36; π(35) = 11; Πd|35 = 35. 1X69. X 2 2 X 36. 2 × 3 × 7; A8; SF24; C24; τ(36) = 8; φ(36) = 10; 4 . 2×25; C35; SF30; D39; PSK17; 3 +7 ; τ(4 ) = 4; σ(36) = 80; π(36) = 11; Πd|36 = 106 0900. φ(4X) = 24; σ(4X) = 76; π(4X) = 14; Πd|4X = 1E44. 37. p12; SF25; D29; τ(37) = 2; φ(37) = 36; σ(37) = E E E X E 38; π(37) = 12; Πd|37 = 37. 4 . p15; SF31; D3X; τ(4 ) = 2; φ(4 ) = 4 ; σ(4 ) = 50; π(4E) = 15; Πd|4E = 4E. 2 38. 2 × E; C25; D2X; L1 + L2 + L3 + L4 + L5 + L6; 22 × 3 × 5 HC C CO A τ(38) = 6; φ(38) = 18; σ(38) = 70; π(38) = 12; 50. ; 8; 36; 17; 10; Sexagesimal τ(50) = 10 φ(50) = 14 σ(50) = 120 Πd|38 = 4 1368. Base; ; ; ; π(50) = 15; Πd|50 = 90 6100 0000. 2 2 2 39. 3 ×5; C26; D2E; 3 +6 ; T9; L1 + L3 + L5 + L7; 2 2 51. p16; SF32; D3E; 5 + 6 ; τ(51) = 2; φ(51) = 50; τ(39) = 6; φ(39) = 20; σ(39) = 66; π(39) = 12; σ(51) = 52; π(51) = 16; Πd|51 = 51. Πd|39 = 4 4899.

52. 2 × 27; C37; SF33; D40; PSK18; τ(52) = 4; 3X. 2 × 1E; C27; D30; SF26; PSK13; τ(3X) = 4; φ(52) = 26; σ(52) = 80; π(52) = 16; Πd|52 = φ(3X) = 1X; σ(3X) = 60; π(3X) = 12; Πd|3X = 2284. 1284. 2 53. 3 × 7; C38; D41; τ(53) = 6; φ(53) = 30; E p D SF L τ(3E) = 2 φ(3E) = 3X 3 . 13; 31; 27; 8; ; ; σ(53) = 88; π(53) = 16; Πd|53 = 10 0853. σ(3E) = 40; π(3E) = 13; Πd|3E = 3E. 6 3 2 5 5 54. 2 ; 4 ; 8 ; 2 +2 ; C39; CO18; D42; S8; τ(54) = 7; 4 X 40. 2 × 3; C28; A9; HC7; CO15; τ(40) = ; φ(40) = φ(54) = 28; σ(54) = X7; π(54) = 16; Πd|54 = X 14; σ(40) = 4; π(40) = 13; Πd|40 = 7140 0000. 85 1768. 72 C D S τ(41) = 3 φ(41) = 36 2 2 2 2 41. ; 29; 32; 7; ; ; 55. 5 × 11; C3X; SF34; D43; PSK19; 1 + 8 ; 4 + 7 ; σ(41) = 49; π(41) = 13; Πd|41 = 247. 13 + 43; 16 + 26; τ(55) = 4; φ(55) = 40; 2 2 2 2 2 2 2 2 σ(55) = 70; π(55) = 16; Πd|55 = 2541. 42. 2 × 5 ; 5 + 5 ; 1 + a7 ; 3 + 4 + 5 ; C2X; D33; τ(42) = 6; φ(42) = 18; σ(42) = 79; π(42) = 13; 56. 2×3×E; C3E; A11; SF35; TE; τ(56) = 8; φ(56) = Πd|42 = 6 0408. 18; σ(56) = 100; π(56) = 16; Πd|56 = 643 0900.

43. 3 × 15; C2E; SF28; D34; PSK14; CO16; τ(43) = 4; 57. p17; SF36; D44; τ(57) = 2; φ(57) = 56; σ(57) = φ(43) = 28; σ(43) = 60; π(43) = 13; Πd|43 = 58; π(57) = 17; Πd|57 = 57. 1609. 2 2 2 58. 2 × 15; C40; 2 + 8 ; D45; CO19; τ(58) = 6; 2 2 2 44. 2 ×11; C30; D35; 4 +6 ; τ(44) = 6; φ(44) = 20; φ(58) = 28; σ(58) = X6; π(58) = 17; Πd|58 = σ(44) = 82; π(44) = 13; Πd|44 = 6 9454. 13 1X68.

6 2 2 2 2 59. 3 × 1E; C41; SF37; D46; PSK1X; τ(59) = 4; 71. 5 × 15; C51; SF45; D55; PSK22; 6 + 7 ; 2 + 9 ; φ(59) = 38; σ(59) = 80; π(59) = 17; Πd|59 = CO1E; τ(71) = 4; φ(71) = 54; σ(71) = 90; 2909. π(71) = 1E; Πd|71 = 4221.

2 2 2 2 5X. 2 × 5 × 7; C42; SF38; A12; τ(5X) = 8; φ(5X) = 20; 72. 2 × 37; C52; SF46; D56; PSK23; 3 + 4 + 5 + 6 ; σ(5X) = 100; π(5X) = 17; Πd|5X = 805 X814. τ(72) = 4; φ(72) = 36; σ(72) = E0; π(72) = 1E; Πd|72 = 4344. 5E. p18; SF39; D47; τ(5E) = 2; φ(5E) = 5X; σ(5E) = 60; π(5E) = 18; Πd|5E = 5E. 73. 3 × 25; C53; SF47; D57; PSK24; τ(73) = 4; φ(73) = 48; σ(73) = X0; π(73) = 1E; Πd|73 = 3 2 2 2 3 3 60. 2 × 3 ; C43; A13; 6 + 6 ; 2 + 4 ; τ(60) = 10; 4469. φ(60) = 20; σ(60) = 143; π(60) = 18; Πd|60 = 3 230 0000 0000. 74. 2 × E; C54; A17; F1 + F2 + F3 + F4 + F5 + F6 + F7 + F8 + F9; F2 + F4 + F6 + F8 2 2 61. p19; SF3X; D48; 3 +8 ; L1 + L2 + L3 + L4 + L5 + FX; τ(74) = 8; φ(74) = 34; σ(74) = 130; + L6 + L7; τ(61) = 2; φ(61) = 60; σ(61) = 62; π(74) = 1E; Πd|74 = 1810 0714. π(61) = 19; Πd|61 = 61. 2 2 75. p20; SF48; D58; FE; 5 + 8 ; τ(75) = 2; φ(75) = 2 2 62. 2×31; C44; SF3E; D49; PSK1E; 5 +7 ; τ(62) = 4; 74; σ(75) = 76; π(75) = 20; Πd|75 = 75. φ(62) = 30; σ(62) = 96; π(62) = 19; Πd|62 = 2 2 2 2 2 2 2 2 3204. 76. 2×3 ×5; C55; A18; 3 +9 ; 2 +3 +4 +5 +6 ; τ(76) = 10; φ(76) = 20; σ(76) = 176; π(76) = 2 63. 3 × 5 ; C45; D4X; L2 + L4 + L6 + L8; τ(63) = 6; 20; Πd|76 = 86EE662 3000. φ(63) = 34; σ(63) = X4; π(63) = 19; Πd|63 = 3 3 2 2 18 4183. 77. 7 × 11; C56; SF49; D59; PSK25; 3 + 4 ; 1 + 2 + 2 2 2 2 3 + 4 + 5 + 6 ; T11; τ(77) = 4; φ(77) = 60; 2 64. 2 × 17; C46; D4E; L9; τ(64) = 6; φ(64) = 30; σ(77) = 94; π(77) = 20; Πd|77 = 4961. σ(64) = E8; π(64) = 19; Πd|64 = 19 2054. 2 78. 2 × 1E; aC57; D5X; τ(78) = 6; φ(78) = 38; 2 2 2 65. 7 × E; C47; D50; SF40; PSK20; 4 + 5 + 6 ; σ(78) = 120; π(78) = 20; Πd|78 = 31 6768. τ(65) = 4; φ(65) = 50; σ(65) = 80; π(65) = 19; Πd|65 = 3521. 79. 3 × 27; C58; SF4X; D5E; PSK26; τ(79) = 4; φ(79) = 50; σ(79) = X8; π(79) = 20; Πd|79 = 66. 2 × 3 × 11; C48; A14; SF41; T10; τ(66) = 8; 5009. φ(66) = 20; σ(66) = 120; π(66) = 19; Πd|66 = 1049 0900. 7X. 2 × 3E; C59; SF4E; D60; PSK27; τ(7X) = 4; φ(7X) = 3X; σ(7X) = 100; π(7X) = 20; Πd|7X = 67. p1X; SF42; D51; τ(67) = 2; φ(67) = 66; σ(67) = 5144. 68; π(67) = 1X; Πd|67 = 67. 7E. 5 × 17; C5X; SF50; D61; PSK28; τ(7E) = 4; 4 2 2 68. 2 × 5; C49; A15; CO1X; 4 + 8 ; τ(68) = X; φ(7E) = 60; σ(7E) = X0; π(7E) = 20; Πd|7E = φ(68) = 28; σ(68) = 136; π(68) = 1X; Πd|68 = 5281. 7 7548 8368. 5 80. 2 × 3; C5E; A19; CO20; τ(80) = 10; φ(80) = 4 2 69. 3 ; 9 ; S9; C4X; D62; τ(69) = 5; φ(69) = 46; 28; σ(80) = 190; π(80) = 20; Πd|80 = σ(69) = X1; π(69) = 1X; Πd|69 = 2 X209. 1078 5400 0000.

4 4 2 2 2 2 4 4 6X. 2 × 35; C4E; SF43; D53; PSK21; 1 + 3 ; 1 + 9 ; 81. p21; 4 + 9 ; 2 + 3 ; SF51; D62; τ(81) = 2; τ(6X) = 4; φ(6X) = 34; σ(6X) = X6; π(6X) = 1X; φ(81) = 80; σ(81) = 82; π(81) = 21; Πd|81 = Πd|6X = 3X84. 81.

2 2 2 4 4 4 6E. p1E; SF44; D54; τ(6E) = 2; φ(6E) = 6X; σ(6E) = 82. 2 × 7 ; 7 + 7 ; C60; D63; 1 + 2 + 3 ; τ(82) = 6; 70; π(6E) = 1E; Πd|6E = 6E. φ(82) = 36; σ(82) = 123; π(82) = 21; Πd|82 = 39 4808. 2 2 2 2 2 70. 2 × 3 × 7; C50; A16; TH7; 1 + 3 + 5 + 7 ; 2 3 3 3 τ(70) = 10; φ(70) = 20; σ(70) = 168;p π(70) = 83. 3 ×E; C61; D64; 2 +3 +4 ; τ(83) = 6; φ(83) = 1E; Πd|70 = 581 0100 0000. 50; σ(83) = 110; π(83) = 21; Πd|83 = 3X 9623.

7 2 2 2 2 2 3 3 3 3 2 2 2 84. 2 ×5 ; C62; A1X; 6 +8 ; X ; SX; 1 +2 +3 +4 ; 99. 3 ×11; C72; D74; 6 +9 ; τ(99) = 6; φ(99) = 60; τ(84) = 9; φ(84) = 34; σ(84) = 161; π(84) = σ(99) = 132; π(99) = 26; Πd|99 = 65 2X39. 21; Πd|84 = 2 3XX9 3854. 9X. 2 × 4E; C73; SF62; D75; PSK30; τ(9X) = 4; 2 2 85. p22; SF52; D65; 1 + X ; τ(85) = 2; φ(85) = 84; φ(9X) = 4X; σ(9X) = 130; π(9X) = 26; Πd|9X = σ(85) = 86; π(85) = 22; Πd|85 = 85. 8084.

2 × 3 × 15 C SF A E CO τ(86) = 8 86. ; 63; 53; 1 ; 21; ; 9E. 7 × 17; C74; SF63; D76; PSK31; τ(9E) = 4; φ(86) = 28; σ(86) = 160; π(86) = 22; Πd|86 = φ(9E) = 80; σ(9E) = 100; π(9E) = 26; Πd|9E = 3030 0900. 8241.

87. p23; SF54; D66; τ(87) = 2; φ(87) = 86; σ(87) = 3 X0. 2 × 3 × 5; 5!; C75; A24; MP1 (3-Perfect); T13; 88; π(87) = 23; Πd|87 = 87. CO22; TH8; L1 + L2 + L3 + L4 + L5 + L6 + 3 2 2 L7 + L8; τ(X0) = 14; φ(X0) = 28; σ(X0) = 260; 88. 2 ×11; C64; A20; 2 +X ; τ(88) = 8; φ(88) = 40; π(X0) = 26; Πd|X0 = 295X 6454 0000 0000. σ(88) = 156; π(88) = 23; Πd|88 = 3321 8194. 2 X1. E ; C76; D77; SE; L1 + L3 + L5 + L7 + L9; 89. 3 × 5 × 7; C65; SF55; D67; T12; τ(89) = 8; X X X E X φ(89) = 40; σ(89) = 140; π(89) = 23; Πd|89 = τ( 1) = 3; φ( 1) = 92; σ( 1) = 1; π( 1) = 26; X E 3485 9969. Πd| 1 = 92 .

2 2 X 2×51 C SF D PSK 12 +E2 τ(X2) = 4 8X. 2×45; C66; SF56; D68; PSK29; 5 +9 ; τ(8X) = 4; 2. ; 77; 64; 78; 32; ; ; X X X X φ(8X) = 44; σ(8X) = 116; π(8X) = 23; Πd|8X = φ( 2) = 50; σ( 2) = 136; π( 2) = 26; Πd| 2 = 6604. 8744. X X 8E. p24; SF57; D69; τ(8E) = 2; φ(8E) = 8X; σ(8E) = 3. 3 × 35; C78; SF65; D79; PSK33; LX; τ( 3) = 4; 90; π(8E) = 24; Πd|8E = 8E. φ(X3) = 68; σ(X3) = 120; π(X3) = 26; Πd|X3 = 8909. 2 3 90. 2 × 3 ; C67; A21; τ(90) = 10; φ(90) = 2 30; σ(90) = 1E4; π(90) = 24; Πd|90 = X4. 2 × 27; C79; D7X; τ(X4) = 6; φ(X4) = 50; 2176 6900 0000. σ(X4) = 168; π(X4) = 26; Πd|X4 = 77 E454.

2 X2 3 2 2 2 2 91. p25; SF58; D6X; 3 + ; τ(91) = 2; φ(91) = 90; X5. 5 ; C7X; D7E; PSK34; 5 + X ; 2 + E ; τ(X5) = 3; σ(91) = 92; π(91) = 25; Πd|91 = 91. φ(X5) = 84; σ(X5) = 110; π(X5) = 26; Πd|X5 = 9061 2 2 2 . 92. 2×5×E; C68; SF59; D6E; 5 +6 +7 ; τ(92) = 8; φ(92) = 34 σ(92) = 160 π(92) = 25 Πd|92 = 2 3 3 2 2 2 2 ; ; ; X6. 2 × 3 × 7; C7E; A25; 1 + 5 ; 4 + 5 + 6 + 7 ; 4104 8014. τ(X6) = 10; φ(X6) = 30; σ(X6) = 220; π(X6) = 26; Πd|X6 = 5476 2778 3000. 93. 3 × 31; C69; SF5X; D70; PSK2X; τ(93) = 4; φ(93) = 60; σ(93) = 108; π(93) = 25; Πd|93 = X7. p27; M7; SF66; D80; τ(X7) = 2; φ(X7) = X6; 7169. σ(X7) = X8; π(X7) = 27; Πd|X7 = X7. 4 94. 2 × 7; C6X; A22; τ(94) = X; φ(94) = 40; σ(94) = 7 2 2 3 3 6 6 X8. 2 ; 8 + 8 ; 4 + 4 ; 2 + 2 ; C80; D81; CO23; 188; π(94) = 25; Πd|94 = 34 EX06 8714. τ(X8) = 8; φ(X8) = 54; σ(X8) = 193; π(X8) = 2 2 27; Πd|X8 = 75X9 4714. 95. p26; SF5E; D71; 7 + 8 ; τ(95) = 2; φ(95) = 94; σ(95) = 96; π(95) = 26; Πd|95 = 95. X9. 3 × 37; C81; SF67; D82; PSK35; τ(X9) = 4; φ(X9) = 70 σ(X9) = 128 π(X9) = 27 Πd|X9 = 96. 2×3×17; C6E; SF60; A23; τ(96) = 8; φ(96) = 30; ; ; ; σ(96) = 180; π(96) = 26; Πd|96 = 4869 0900. 9769. XX 2 2 2 E2 97. 5 × 1E; C70; SF61; D72; PSK2E; τ(97) = 4; . 2 × 5 × 11; C82; SF68; D83; 7 + 9 ; 3 + ; φ(97) = 74; σ(97) = 100; π(97) = 26; Πd|97 = τ(XX) = 8; φ(XX) = 40; σ(XX) = 190; π(XX) = 77X1. 27; Πd|XX = 7E79 7694.

2 2 2 98. 2 ×25; C71; D73; 4 +X ; τ(98) = 6; φ(98) = 48; XE. p28; SF69; τ(XE) = 2; φ(XE) = XX; σ(XE) = E0; σ(98) = 156; π(98) = 26; Πd|98 = 63 3368. π(XE) = 28; Πd|XE = XE.

8 2 2 2 2 2 2 2 E0. 2 × 3 × E; C83; A26; τ(E0) = 10; φ(E0) = E7. p2X; SF72; D8X; 2 + 3 + 4 + 5 + 6 + 7 ; 34; σ(E0) = 240; π(E0) = 28; Πd|E0 = τ(E7) = 2; φ(E7) = E6; σ(E7) = E8; π(E7) = 2X; 2 715261000000.2 × 3 × E; C83; A26; τ(E0) = 10; Πd|E7 = E7. φ(E0) = 34; σ(E0) = 240; π(E0) = 28; Πd|E0 = E 2 2 2 2 2 2 2 2 7152 6100 0000. 8. 2 ×5×7; C89; A28; 1 +2 +3 +4 +5 +6 +7 ; τ(E8) = 10; φ(E8) = 40; σ(E8) = 240; π(E8) = E 3 3 E 1. 7×17; C84; SF6X; D85; PSK36; 2 +5 ; τ( 1) = 4; 2X; Πd|E8 = X173 3645 6454. φ(E1) = 90; σ(E1) = 114; π(E1) = 28; Πd|E1 = X X 2 1. E9. 3 × 3E; C8X; SF73; D8E; PSK38; τ(E9) = 4; φ(E9) = 78; σ(E9) = 140; π(E9) = 2X; Πd|E9 = E2. 2 × 57; C85; SF6E; D86; PSK37; τ(E2) = 4; E609 φ(E2) = 56; σ(E2) = 150; π(E2) = 28; Πd|E2 = . X484. EX. 2 × 5E; C8E; SF74; D90; PSK39; τ(EX) = 4; 3 2 2 2 2 2 EX X EX EX X EX E3. 3 ×5; C86; D87; 3 +4 +5 +6 +7 ; τ(E3) = 8; φ( ) = 5 ; σ( ) = 160; π( ) = 2 ; Πd| = φ(E3) = 60; σ(E3) = 180; π(E3) = 28; Πd|E3 = E804. 932X 0969. EE. E × 11; C90; SF75; D91; PSK3X; F1 + F2 + E 3 2 X2 E 4. 2 × 15; C87; D88; CO24; 6 + ; T14; τ( 4) = 8; F3 + F4 + F5 + F6 + F7 + F8 + F9 + E E E E φ( 4) = 54; σ( 4) = 180; π( 4) = 28; Πd| 4 = FX; τ(EE) = 4; φ(EE) = X0; σ(EE) = 120; E 9669 854. π(EE) = 2X; Πd|EE = EX01. E 2 2 5. p29; SF70; D89; 4 + E ; τ(E5) = 2; φ(E5) = E4; 4 2 2 100. 2 × 3 ; One Gross; C91; A29; F10; 10 ; 4! + σ(E5) = E6; π(E5) = 29; Πd|E5 = E5. 5!; 3! + 4!; F1 + F3 + F5 + F7 + F9 + FE; E6. 2×3×1E; C88; SF71; A27; τ(E6) = 8; φ(E6) = 38; S10; τ(100) = 13; φ(100) = 40; σ(100) = 297; σ(E6) = 200; π(E6) = 29; Πd|E6 = X156 0900. π(100) = 2X; Πd|100 = 1000 0000 0000, 0000.

This document was originally published in The Duodecimal τ(9E), and φ(9E) had to be determined by consulting the OEIS, Bulletin 38:3 (WN 74), pp. 9–1E. The original printing in the since the original printing did not include them. Otherwise, Bulletin contained a printing error, which combined the para- however, all the calculations were done by Prof. Jay Schiffman, graphs for 9X and 9E into one, labelled 9X, and did not have a DSA. The Dozenal Society of America is proud to present this separate paragraph for 9E. This has been fixed; this repair re- entirely newly typeset and corrected version to the world, Au- quired a few additions. Specifically, φ(9X), σ(9X), π(9X), Πd|9X, gust 1201.