From the table, it's easy to see that there are 5 x 3 = 15 divisors of 144. In general, if you have the prime factorization of the number n, then to calculate how many divisors it has, you take all the exponents in the factorization, add 1 to each, and then multiply these 'exponents + 1's together. These division worksheets can be configured to layout the division problems using the division sign or a slash (/) format. You may select between 12 and 30 problems for these division worksheets. Negative Number Division Worksheets Horizontal Format These division worksheets may be configured for the divisors and quotients in the range of -12. If I understand correctly, you want all the divisors for each number in the range 1.,1000000? If this is so, why are you checking all the possible divisors for each number in the range? Whatever is the list of divisors for 10 is also in the list for 20, 30,., 90, 100,., 1000 etc. – shapiro yaacov Oct 11 '15 at 5:57.
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of. The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n / m is again an integer (which is necessarily also a divisor of n ). For example, 3 is a divisor of 21, since 21/7 = 3 (and 7 is also a divisor of 21). IRA Required Minimum Distribution Worksheet Use this worksheet to figure this year’s required withdrawal for your traditional IRA UNLESS your spouse1 is the sole beneficiary of your IRA and he or she is more than 10 years younger than you. Deadline for receiving required minimum distribution.
Divisor function σ0(n) up to n = 250
Sigma function σ1(n) up to n = 250
Sum of the squares of divisors, σ2(n), up to n = 250
Sum of cubes of divisors, σ3(n) up to n = 250
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
- 4Properties
Divisors Of 13
Definition[edit]
The sum of positive divisors function σx(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as
where is shorthand for 'ddividesn'.The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function[1][2] (OEIS: A000005). When x is 1, the function is called the sigma function or sum-of-divisors function,[1][3] and the subscript is often omitted, so σ(n) is the same as σ1(n) (OEIS: A000203).
The aliquot sums(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS: A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.
Example[edit]
For example, σ0(12) is the number of the divisors of 12:
while σ1(12) is the sum of all the divisors:
and the aliquot sum s(12) of proper divisors is:
Table of values[edit]
The cases x = 2 to 5 are listed in OEIS: A001157 − OEIS: A001160, x = 6 to 24 are listed in OEIS: A013954 − OEIS: A013972.
n | factorization | σ0(n) | σ1(n) | σ2(n) | σ3(n) | σ4(n) |
---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 2 | 2 | 3 | 5 | 9 | 17 |
3 | 3 | 2 | 4 | 10 | 28 | 82 |
4 | 22 | 3 | 7 | 21 | 73 | 273 |
5 | 5 | 2 | 6 | 26 | 126 | 626 |
6 | 2‧3 | 4 | 12 | 50 | 252 | 1394 |
7 | 7 | 2 | 8 | 50 | 344 | 2402 |
8 | 23 | 4 | 15 | 85 | 585 | 4369 |
9 | 32 | 3 | 13 | 91 | 757 | 6643 |
10 | 2‧5 | 4 | 18 | 130 | 1134 | 10642 |
11 | 11 | 2 | 12 | 122 | 1332 | 14642 |
12 | 22‧3 | 6 | 28 | 210 | 2044 | 22386 |
13 | 13 | 2 | 14 | 170 | 2198 | 28562 |
14 | 2‧7 | 4 | 24 | 250 | 3096 | 40834 |
15 | 3‧5 | 4 | 24 | 260 | 3528 | 51332 |
16 | 24 | 5 | 31 | 341 | 4681 | 69905 |
17 | 17 | 2 | 18 | 290 | 4914 | 83522 |
18 | 2‧32 | 6 | 39 | 455 | 6813 | 112931 |
19 | 19 | 2 | 20 | 362 | 6860 | 130322 |
20 | 22‧5 | 6 | 42 | 546 | 9198 | 170898 |
21 | 3‧7 | 4 | 32 | 500 | 9632 | 196964 |
22 | 2‧11 | 4 | 36 | 610 | 11988 | 248914 |
23 | 23 | 2 | 24 | 530 | 12168 | 279842 |
24 | 23‧3 | 8 | 60 | 850 | 16380 | 358258 |
25 | 52 | 3 | 31 | 651 | 15751 | 391251 |
26 | 2‧13 | 4 | 42 | 850 | 19782 | 485554 |
27 | 33 | 4 | 40 | 820 | 20440 | 538084 |
28 | 22‧7 | 6 | 56 | 1050 | 25112 | 655746 |
29 | 29 | 2 | 30 | 842 | 24390 | 707282 |
30 | 2‧3‧5 | 8 | 72 | 1300 | 31752 | 872644 |
31 | 31 | 2 | 32 | 962 | 29792 | 923522 |
32 | 25 | 6 | 63 | 1365 | 37449 | 1118481 |
33 | 3‧11 | 4 | 48 | 1220 | 37296 | 1200644 |
34 | 2‧17 | 4 | 54 | 1450 | 44226 | 1419874 |
35 | 5‧7 | 4 | 48 | 1300 | 43344 | 1503652 |
36 | 22‧32 | 9 | 91 | 1911 | 55261 | 1813539 |
37 | 37 | 2 | 38 | 1370 | 50654 | 1874162 |
38 | 2‧19 | 4 | 60 | 1810 | 61740 | 2215474 |
39 | 3‧13 | 4 | 56 | 1700 | 61544 | 2342084 |
40 | 23‧5 | 8 | 90 | 2210 | 73710 | 2734994 |
41 | 41 | 2 | 42 | 1682 | 68922 | 2825762 |
42 | 2‧3‧7 | 8 | 96 | 2500 | 86688 | 3348388 |
43 | 43 | 2 | 44 | 1850 | 79508 | 3418802 |
44 | 22‧11 | 6 | 84 | 2562 | 97236 | 3997266 |
45 | 32‧5 | 6 | 78 | 2366 | 95382 | 4158518 |
46 | 2‧23 | 4 | 72 | 2650 | 109512 | 4757314 |
47 | 47 | 2 | 48 | 2210 | 103824 | 4879682 |
48 | 24‧3 | 10 | 124 | 3410 | 131068 | 5732210 |
49 | 72 | 3 | 57 | 2451 | 117993 | 5767203 |
50 | 2‧52 | 6 | 93 | 3255 | 141759 | 6651267 |
Properties[edit]
Formulas at prime powers[edit]
For a prime numberp,
because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the primorial,
since n prime factors allow a sequence of binary selection ( or 1) from n terms for each proper divisor formed.
Clearly, and σ(n) > n for all n > 2.
The divisor function is multiplicative, but not completely multiplicative:
The consequence of this is that, if we write
Table Of Divisors Worksheet
where r = ω(n) is the number of distinct prime factors of n, pi is the ith prime factor, and ai is the maximum power of pi by which n is divisible, then we have: [4]
which is equivalent to the useful formula: [4]
It follows (by setting x = 0) that d(n) is: [4]
For example, if n is 24, there are two prime factors (p1 is 2; p2 is 3); noting that 24 is the product of 23×31, a1 is 3 and a2 is 1. Thus we can calculate as so:
The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
Other properties and identities[edit]
Euler proved the remarkable recurrence:[5][6]
where we set if it occurs and for , we use the Kronecker delta and are the pentagonal numbers. Indeed, Euler proved this by logarithmic differentiation of the identity in his Pentagonal number theorem.
For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and is odd. Similarly, the number is odd if and only if n is a square or twice a square.[citation needed]
We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, that is, the divisors of n excluding n itself. This function is the one used to recognize perfect numbers which are the n for which s(n) = n. If s(n) > n then n is an abundant number and if s(n) < n then n is a deficient number.
If n is a power of 2, for example, , then and s(n) = n - 1, which makes nalmost-perfect.
As an example, for two distinct primes p and q with p < q, let
Then
and
where is Euler's totient function.
Then, the roots of:
allow us to express p and q in terms of σ(n) and φ(n) only, without even knowing n or p+q, as:
Also, knowing n and either or (or knowing p+q and either or ) allows us to easily find p and q.
In 1984, Roger Heath-Brown proved that the equality
is true for an infinity of values of n, see OEIS: A005237.
Series relations[edit]
Two Dirichlet series involving the divisor function are: [7]
which for d(n) = σ0(n) gives: [7]
and [8]
A Lambert series involving the divisor function is: [9]
for arbitrary complex |q| ≤ 1 and a. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.
For exists an explicit series representation with Ramanujan sums as :[10]
The computation of the first terms of shows its oscillations around the 'average value' :
Growth rate[edit]
In little-o notation, the divisor function satisfies the inequality:[11][12]
More precisely, Severin Wigert showed that:[12]
On the other hand, since there are infinitely many prime numbers,[12]
In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:[13][14]
where is Euler's gamma constant. Improving the bound in this formula is known as Dirichlet's divisor problem.
The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: [15]
where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that:
where p denotes a prime.
In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:
- (Robin's inequality)
holds for all sufficiently large n (Ramanujan 1997). The largest known value that violates the inequality is n=5040. In 1984, Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n > 5040 must be superabundant (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime (Choie et al. 2007).
Robin also proved, unconditionally, that the inequality:
holds for all n ≥ 3.
A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:
for every natural numbern > 1, where is the nth harmonic number, (Lagarias 2002).
See also[edit]
- Divisor sum convolutions Lists a few identities involving the divisor functions
- Euler's totient function (Euler's phi function)
Notes[edit]
- ^ abLong (1972, p. 46)
- ^Pettofrezzo & Byrkit (1970, p. 63)
- ^Pettofrezzo & Byrkit (1970, p. 58)
- ^ abcHardy & Wright (2008), pp. 310 f, §16.7.
- ^Euler, Leonhard; Bell, Jordan (2004). 'An observation on the sums of divisors'. arXiv:math/0411587.
- ^http://eulerarchive.maa.org//pages/E175.html, Decouverte d'une loi tout extraordinaire des nombres par rapport a la somme de leurs diviseurs
- ^ abHardy & Wright (2008), pp. 326-328, §17.5.
- ^Hardy & Wright (2008), pp. 334-337, §17.8.
- ^Hardy & Wright (2008), pp. 338-341, §17.10.
- ^E. Krätzel (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130. (German)
- ^Apostol (1976), p. 296.
- ^ abcHardy & Wright (2008), pp. 342-347, §18.1.
- ^Apostol (1976), Theorem 3.3.
- ^Hardy & Wright (2008), pp. 347-350, §18.2.
- ^Hardy & Wright (2008), pp. 469-471, §22.9.
References[edit]
- Akbary, Amir; Friggstad, Zachary (2009), 'Superabundant numbers and the Riemann hypothesis'(PDF), American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128, archived from the original(PDF) on 2014-04-11.
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN978-0-387-90163-3, MR0434929, Zbl0335.10001
- Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN0-262-02405-5, see page 234 in section 8.8.
- Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011), 'Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis'(PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 11: A33, arXiv:1110.5078, Bibcode:2011arXiv1110.5078C
- Choie, YoungJu; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), 'On Robin's criterion for the Riemann hypothesis', Journal de théorie des nombres de Bordeaux, 19 (2): 357–372, arXiv:math.NT/0604314, doi:10.5802/jtnb.591, ISSN1246-7405, MR2394891, Zbl1163.11059
- Grönwall, Thomas Hakon (1913), 'Some asymptotic expressions in the theory of numbers', Transactions of the American Mathematical Society, 14: 113–122, doi:10.1090/S0002-9947-1913-1500940-6
- Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN978-0-19-921986-5, MR2445243, Zbl1159.11001
- Ivić, Aleksandar (1985), The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York etc.: John Wiley & Sons, pp. 385–440, ISBN0-471-80634-X, Zbl0556.10026
- Lagarias, Jeffrey C. (2002), 'An elementary problem equivalent to the Riemann hypothesis', The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN0002-9890, JSTOR2695443, MR1908008
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN77171950
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN77081766
- Ramanujan, Srinivasa (1997), 'Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin', The Ramanujan Journal, 1 (2): 119–153, doi:10.1023/A:1009764017495, ISSN1382-4090, MR1606180
- Robin, Guy (1984), 'Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann', Journal de Mathématiques Pures et Appliquées, Neuvième Série, 63 (2): 187–213, ISSN0021-7824, MR0774171
- Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge: Cambridge University Press, ISBN978-0-521-17562-3, Zbl1227.11002
External links[edit]
- Weisstein, Eric W.'Divisor Function'. MathWorld.
- Weisstein, Eric W.'Robin's Theorem'. MathWorld.
- Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.
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