On Some Functions Involving the lcm and gcd of Integer Tuples

Transcription

1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact: In this pape an explicit fomula fo the numbe of tuples of positive integes having the same lowest common multiple n is deived, and some of the popeties of the esulting aithmetic function ae analyzed. The tuples having also the same geatest common diviso ae investigated, while some novel o existing intege sequences ae ecoveed as paticula cases. A fomula linking the gcd and lcm fo tuples of integes is also pesented. Keywods: divisibility, lowest common multiple, geatest common diviso, intege sequences 1 Intoduction The least common multiple of two natual numbes a and b, is usually denoted by lcm(a,b) o a,b, and is the smallest numbe divisible by both a and b 5, 5.1, p.48. The dual notion is the geatest common diviso, denoted by gcd(a,b) o (a,b) which is the lagest numbe that divides both a and b. The lcm and gcd can also be defined fo any k-tuple of natual numbes a 1,...,a k, whee k 2. Fo n = p n 1, the numbe of odeed pais (a,b) having the same lcm n is {(a,b) : lcm(a,b) = n} = (2n 1 + 1)(2n 2 + 1) (2n + 1), (1) whee is the cadinality of a set 10. If n is squae-fee we have 3 ω(n) 4, ex. 2.4, p.101), whee ω(n) denotes the numbe of pime divisos of n. The numbe of elatively pime odeed pais (a,b) having the same lcm n is (see 8) {(a,b) : gcd(a,b) = 1, lcm(a,b) = n} = 2 ω(n). (2) A popety linking the lcm and gcd of the intege pai (a,b) is (see 5) gcd(a,b)lcm(a,b) = ab. (3) The aim of this pape is to extend the above esults to geneal tuples of integes. Manuscipt eceived Mach 21 ; accepted June 28 O. Bagdasa is with the School of Computing and Mathematics, Univesity of Deby, Kedleston Road, Deby DE22 1GB, England, U.K., o.bagdasa deby.ac.uk 91

2 92 O. Bagdasa The numbe of k-tuples of positive integes with the least common multiple n is LCM(n;k) = {(a 1,...,a k ) : lcm(a 1,...,a k ) = n}, (4) and some identities and inequalities involving the above aithmetic function ae pesented. A few sequences indexed in the On-Line Encyclopedia of Intege Sequences (OEIS) 7, ae obtained as paticula cases. The numbe of odeed k-tuples with the same gcd d and lcm n is defined by GL(d,n;k) = {(a 1,...,a k ) : gcd(a 1,...,a k ) = d, lcm(a 1,...,a k ) = n}, (5) and a numbe of popeties of this function ae analyzed. In the pocess, some families of intege sequences not cuently indexed in the OEIS ae poduced. Finally, a popety linking the lcm and gcd fo k-tuples of integes which genealizes fomula (3) is pesented, accompanied by a bief poof based on 9. 2 Main esults In this section the poofs fo the fomulae of LCM and GL ae povided, along with a esult linking the lcm and gcd computed fo k-tuples of integes. 2.1 Tuples of integes with the same lcm This section pesents the fomula fo the numbe of odeed k-tuples whose least common multiple is a natual numbe n. Some of the popeties of this function ae then investigated. Theoem 2.1 Let k and n be natuals numbes. If n has the factoization n = p n 1 the numbe of odeed k-tuples whose lcm is n is given by the fomula LCM(n;k) =, (n i + 1) k n k i. (6) Poof. Let {a 1,...,a k } be a k-tuple satisfying a 1,...,a k = n. The numbes a 1,...,a k need to be divisos of n, theefoe they can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, (7) with 0 a i j n i (i = 1,...,). Because thei lcm is n, one needs to have max{a i1,...,a ik } = n i, (8) fo each diviso p i, i = 1,...,. Fo each i {1,...,}, the total numbe of pais 0 a i j n i ( j = 1,...,k) is (n i + 1) k. On the othe hand, the numbe of odeed pais 0 a i j n i 1 ( j = 1,...,k) not having n i as a maximum is n k i. The numbe of pais satisfying elation (8) is theefoe (n i + 1) k n k i. Multiplying this fomula fo i = 1,..., one obtains (6).

3 On some functions involving the lcm and gcd of intege k-tuples 93 Remak 2.1 Fo paticula values of k, one ecoves the following OEIS indexed intege sequences: A fo LCM(n; 2) (linked to numeous algebaic intepetations), and A070919, A070920, A fo LCM(n;3), LCM(n;4), LCM(n;5), espectively. The following esult illustates the independence of LCM(n;k) on the pime factos. Coollay 2.1 Let k be a natual numbe and n = p n 1, m = q n 1 1 qn qn, such that all numbes p 1,..., p,q 1,...,q ae distinct. Then LCM(m;k) = LCM(n;k). Poof. Fom (6), LCM(n; k) only depends on the multiplicities of the pime factos, and not on the pime factos themselves. In this case LCM(m;k) = (n i + 1) k n k i = LCM(n;k) Consideing elatively pime numbes m and n, one obtains the consequence below, which hee is poved using a diect method. Coollay 2.2 Let m,n be integes satisfying (m,n) = 1. The following popety holds LCM(m n;k) = LCM(m;k) LCM(n;k). (9) Poof. Conside the following factoizations of m = q m 1 1 qm qm s s and n = p n 1. As (m,n) = 1, it follows that p i q j, fo any combination of i = 1,..., and j = 1,...,s. This implies that the pime factoization of the poduct mn is m n = q m 1 1 qm 2 2 qm s s p n 1 2 pn, with the pime factos not necessaily in an inceasing ode. By fomula (6) one obtains LCM(m n;k) = (m i + 1) k m k s i (n j + 1) k n k j = LCM(m;k) LCM(n;k), j=1 which ends the poof. Hee ae ecalled some well known popeties of aithmetic functions 1, Chapte 2. An aithmetic function f (n) of the positive intege n is multiplicative if f (1) = 1 and fo any a and b copime, then f (ab) = f (a) f (b). An aithmetic function is said completely multiplicative if f (1) = 1 and f (ab) = f (a) f (b), even when a and b ae not copime. Remak 2.2 Fom Coollay 2.2, the function LCM(n;k) is multiplicative. Remak 2.3 Let a,b be positive integes and f : N N be a multiplicative aithmetic function. The following popety holds f (gcd(a,b)) f (lcm(a,b)) = f (a) f (b). (10) Poof. Let d = gcd(a,b). The numbes d, a/d and b/d ae copime. Using (3) one obtains f (d) f (lcm(a,b)) = f (d) f (d) f (a/d) f (b/d) = f (d) f (a/d) f (d) f (b/d) = f (a) f (b). This ends the poof.

4 94 O. Bagdasa Fo LCM(n;k) we can check diectly the popety elative to lcm and gcd. Coollay 2.3 Let a,b be natual numbes. The following popety holds: LCM(gcd(a,b);k) LCM(lcm(a,b);k) = LCM(a;k) LCM(b;k). (11) Poof. Assume that the factoizations of n,d,a,b ae n = p n 1, d = p d 1 1 pd pd s and a = p α 1 1 pα pα, b = p β 1 1 pβ pβ, whee some of the powes may be zeo. Using the elation (6), it is sufficient to pove the fomula fo just one fixed value i fom 1,...,. The contibution of the pime facto p i to fomula (11) is (n i + 1) k n k i (d i + 1) k di k = (α i + 1) k α k i This elation is tue, as d i = min{α i,β i } and n i = max{α i,β i }. (β i + 1) k βi k Some inequalities fo LCM(n;k) can also be poved, fo geneal values of m and n. Theoem 2.2 Let k, m and n be natual numbes. The following inequality holds LCM(m n;k) LCM(m;k) LCM(n;k) (12) To pove this theoem, we shall fist pove the following esult. Lemma 2.3 Let k 2, p be a pime numbe and α,β 1 natual numbes. Then LCM(p α+β ;k) LCM(p α ;k) LCM(p β ;k) (13) Poof. The inequality (13) is then equivalent to (α + β + 1) k (α + β) k (α + 1) k α k (β + 1) k β k. (14) Fom (6) we have LCM(p α+β ;k) = (α + β + 1) k (α + β) k = k 1 ) (α + β) l, then ( k l=0 l LCM(p α ;k) = (α +1) k α k = i=0 k 1 ( k ) i α i and LCM(p β ;k) = (β +1) k β k = k 1 ( k j=0 j) β j. Fo a fixed pai (i, j) with i, j chosen fom the set {0,...,k 1}, the tem α i β j appeas in the ight-hand side, and its coefficient is ( k k i)( j). On the othe hand, the tem α i β j only appeas on the left-hand side if l = i + j k 1, and its coefficient is ( k l)( l i). To finalize the poof it emains to be shown that the latte coefficient is smalle than the fome. One can check that ( k )( k i j) k(k 1) (k i + 1) ( k )( l = l i) (k j)(k j 1) (k j i + 1) 1. The computational details of the poof ae left to the eade as an execise. Conside the factoizations of m = p m 1 1 pm pm and n = p n 1, whee m i,n i 0. Using Coollay 2.2 fo the pime factos p 1,..., p and thei powes, and Lemma 2.3 fo each of the indices i = 1,...,, one obtains (12). This ends the poof of the theoem. Remak 2.4 It is not difficult to pove that using paticula values that the inequality in Theoem 2.2 is stict fo k 2. Fo example, fo p = 2, α = β = 1 one obtains the elation 3 k 2 k < (2 k 1 k )(2 k 1 k ), which is tue fo all values k 2. This also poves that LCM(n;k) is not completely multiplicative.

5 On some functions involving the lcm and gcd of intege k-tuples Tuples of integes with the same lcm and gcd This section evaluates the function GL(d, n; k) given by (5), enumeating the k-tuples with fixed geatest common diviso d and least common multiple n. Some of its popeties and paticula instances ae also pesented, along with some novel numbe sequences. The following lemma epesents the motivation fo the esults in this section. Lemma 2.4 Let d < n be positive integes, such that d n. The numbe of odeed pais (a,b) with the same geatest common diviso d and least common multiple n is {(a,b) : gcd(a,b) = d, lcm(a,b) = n} = 2 ω(n/d). (15) whee ω(x) epesents the numbe of distinct pime divisos fo the intege x. Poof. Let the numbes d,n,a,b have the pime decompositions d = p d 1 1 pd 2 2 pd, n = p n 1 2 pn, a = p α 1 1 pα 2 2 pα, b = p β 1 1 pβ 2 2 pβ, whee 1 d i n i fo i = 1,...,. As d = (a,b) and n = a,b, d i = min{α i,β i } and n i = max{α i,β i } fo each i {1,...,}. Also, the numbes a and b ae distinct, o othewise d = n. Thee ae two possibilities. When d i = n i, one has α i = β i = d i = n i. Each choice of i I = {i {1,...,n} : d i < n i } geneates two possible pais (α i,β i ) {(d i,n i ),(n i,d i )}, hence in total thee ae 2 I distinct pais of powes. As the pime decomposition of n/d is n/d = p n 1 d 1 1 p n 2 d 2 2 p n d = p n i d i i, i I one obtains that I = ω(n/d). This ends the poof. Fo d = 1 one ecoves the esult (2), epesenting the numbe of unitay divisos, indexed as A in the OEIS. The following esult elates GL(d,n;k) to the set of elatively pime tuples, and shows that finding GL(d,n;k) can be educed to evaluating GL(1,n/d;k). Lemma 2.5 Let k and d n be natual numbes. If d = p d 1 1 pd pd and n = p n 1 the numbe of odeed k-tuples whose gcd is d, and lcm is n satisfies the popety GL(d,n;k) = GL(1,n/d;k). (16) Poof. Let {a 1,...,a k } be a k-tuple satisfying (a 1,...,a k ) = d and a 1,...,a k = n. Numbes a 1,...,a k ae multiples of d and divisos of n, theefoe they can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, with d i a i j n i (i = 1,...,). Because thei gcd is d and lcm is n, one needs to have min{a i1,...,a ik } = d i, max{a i1,...,a ik } = n i, (17),

6 96 O. Bagdasa fo each p i, i = 1,...,. The numbe of tuples in (17) is the same as of the tuples satisfying min{a i1,...,a ik} = 0, max{a i1,...,a ik} = n i d i, which elate to the k-tuples having the popeties (a 1,...,a k ) = 1 and a 1,...,a k = n/d. This ends the poof. Lemma 2.6 Let k and α be positive integes. The numbe of tuples (α 1,...,α k ) satisfying T(α;k) = {(α 1,...,α k ) : min(α 1,...,α k ) = 0, max(α 1,...,α k ) = α} (18) is given by the fomula T(α;k) = (α + 1) k 2α k + (α 1) k. (19) Poof. Conside the tem T(α;k+1) epesenting the numbe of (k+1)-tuples (α 1,...,α k+1 ) whose min is 0 and max is α. Thee ae thee distinct cases, depending on the value of α k α k+1 = 0. In this case the k-tuple (α 1,...,α k ) has to satisfy max(α 1,...,α k ) = α. Following the agument used in Theoem 2.1, the numbe of k-tuples having this popety is (α + 1) k α k. 2. α k+1 = n. Hee the estiction on the k-tuple (α 1,...,α k ) is min(α 1,...,α k ) = 0. Similaly as above, the numbe of k-tuples with this popety is (α + 1) k α k < α k+1 < n. Fo each of the (α 1) possible values of α k+1, the k-tuples (α 1,...,α k ) satisfy min(α 1,...,α k ) = 0 and max(α 1,...,α k ) = α, and thei numbe is T(α;k). By counting all the (k + 1)-tuples fo a fixed α, the tems T(α;k) satisfy the ecuence T(α;k + 1) = 2 (α + 1) k α k + (α 1)T(α;k) (20) Fo α 1 one has T(α;0) = T(α;1) = 0, while fom the poof of Lemma 2.5, T(α;2) = 2. Witing the ecusion until T(α;1), one obtains T(α;k + 1) = 2 = 2 k i=0 k i=0 (α 1) i (α + 1) k i α k i + (α 1) k+1 T (α;0) (α 1) i (α + 1) k i (α 1) i α k i (α + 1) k+1 α k+1 = 2 2 αk+1 (α 1) k+1 1 = (α + 1) k+1 2α k+1 + (α 1) k+1, whee the elation a k+1 b k+1 = (a b) ( a k + a k 1 b + + ab k 1 + b k) valid fo any a,b was used. The esult poves (19), which could also be checked by mathematical induction. Hee the diect poof was pefeed, as it shows how the fomula can be deived.

7 On some functions involving the lcm and gcd of intege k-tuples 97 Fo a simplified notation we may define L(n;k) := GL(1,n;k), fo positive integes k,n. The fomula fo L(n;k) now follows as a diect consequence of Lemma 2.6. Theoem 2.7 Let k and n be natuals numbes. If n has the factoization n = p n 1 the numbe of odeed k-tuples whose gcd is 1 and lcm is n, is given by the fomula L(n;k) = (n i + 1) k 2n k i + (n i 1) k. (21) Poof. Let {a 1,...,a k } be a k-tuple satisfying gcd(a 1,...,a k ) = 1 and lcma 1,...,a k = n. The numbes a 1,...,a k can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, with 0 a i j n i (i = 1,...,). Because thei gcd is 1 and lcm is n, one needs to have min{a i1,...,a ik } = 0, max{a i1,...,a ik } = n i, fo each p i, i = 1,...,, which fom Lemma 2.6 poduces (n i + 1) k 2n k i +(n i 1) k tuples. By taking the poduct ove all factos p i, i = 1,..., one obtains (21). The following popeties of L(n; k) ae diect consequences of fomula (21), simila to the esults obtained fo LCM(n;k). Fo this eason these ae pesented hee without poofs. The fist popety suggests the exclusive dependence of L(n;k) on the pime factoization. Coollay 2.4 Let k be a natual numbe and n = p n 1, m = q n 1 1 qn qn, such that all numbes p 1,..., p,q 1,...,q ae distinct. Then L(m;k) = L(n;k). The following esults states the multiplicity of the aithmetic function L(n;k) fo k 2. Coollay 2.5 Let m,n be compime integes and k 2. The following popety holds The last esult is a consequence of the multiplicity. L(m n;k) = L(m;k) L(n;k). (22) Coollay 2.6 Let a,b be natual numbes. The following popety holds: L(gcd(a,b);k) L(lcm(a,b);k) = L(a;k) L(b;k). (23) Remak 2.5 Choosing m = n = 2 and k 2 one may check that L(m n;k) = 3 k 2 2 k + 1 < (2 k 2) (2 k 2) = L(m;k) L(n;k). This poves that L(n;k) is not completely multiplicative. Peliminay investigations suggest that the inequality L(m n; k) L(m; k) L(n; k) is tue. Howeve, a complete solution is not known to the autho.,

8 98 O. Bagdasa 2.3 Intege sequences elated to T(n;k) and L(n;k) A numbe of intege sequences can be obtained fom T(n;k), fo paticula values of k and n. These novel, o existing OEIS numbe sequences have a unified intepetation, epesenting the numbe of odeed k-tuples whose minimum is zeo and maximum is n. Fo n = 1, one obtains T(1;k) = 2 k 2, which is the numbe of nonempty pope subsets fo a set with k elements, indexed as A in the OEIS. Fo n = 2, the sequence defined by T(2;k 1) = 3 k 1 2 k + 1, epesents the Stiling numbes of the second kind, indexed as A in the OEIS. The fist few tems of the sequences T(n;k) fo n = 3,4,5 give T(3;k) : 0,2,18,110,570,2702,12138,52670,223290,931502,... (24) T(4;k) : 0,2,24,194,1320,8162,47544,266114, , ,... (25) T(5;k) : 0,2,30,302,2550,19502,140070,963902, , ,... (26) not cuently indexed in the OEIS, suggesting that they ae pobably new numbe sequences. Fo k = 1,2,3, one obtains T(n;1) = 0, T(n;2) = 2 and T(n;1) = 3n, espectively. Fo k = 4, the sequence defined by T(n;4) = 12n 2 + 2, epesents the numbe of points on suface of hexagonal pism, indexed as A in the OEIS. Finally, fo k = 5 one obtains T(n;5) = 20n n, which is A (also, T(n + 1;5) poduces A101098). Seveal intege sequences ae obtained fom L(n; k), fo paticula values of k and n. Some special cases woth mentioning. Fo a pime numbe p, one has L(p;k) = 2 k 2. Also, the values L(n;1) = 0 and L(1;k) = 1 ae poduced, fo a positive intege n and k 2. If n = p n 1, the following esults ae obtained as paticula cases: L(n; 2) = (n i + 1) 2 2n 2 i + (n i 1) 2 = 2 ω(n), (27) L(n; 3) = L(n; 4) = L(n; 5) = (n i + 1) 3 2n 3 i + (n i 1) 3 = 6 ω(n) (n i + 1) 4 2n 4 i + (n i 1) 4 = 2 ω(n) (n i + 1) 5 2n 5 i + (n i 1) 5 = 10 ω(n) n i, (28) 6n 2 i + 1, (29) 2n 3 i + n i. (30) Fo some paticula values of k, a numbe of novel intege sequences can be ecoveed. Fo k = 2, one obtains L(n;2) = 2 ω(n), which is as seen befoe, is A in the OEIS. The fist few tems of the numbe sequences L(n;k) fo k = 3,4,5 give L(n;3) : 1,6,6,12,6,36,6,18,12,36,6,72,6,36,36,... (31) L(n;4) : 1,14,14,50,14,196,14,110,50,196,14,700,14,196,196,... (32) L(n;5) : 1,30,30,180,30,900,30,570,180,900,30,5400,30,900,900,... (33) not cuently indexed in the OEIS. The enumeation of esults in this section suggests that some of the sequences pesented ae new. Also, some existing sequences wee given a diffeent intepetation.

9 On some functions involving the lcm and gcd of intege k-tuples A link between the lcm and gcd of k-tuples of integes One can use the pime numbe factoization to pove the following elations, which link the lcm and gcd of k-tuples of integes. Detailed poofs can be found in 9. Theoem 2.8 Let k 2 and a 1,...,a k be natual numbes. The following popety holds gcd(a i1,...,a iu ) 1 i lcm(a 1,a 2,...,a k ) = 1 <...<i u k gcd(a i1,...,a iv ), (34) 1 i 1 <...<i v k whee u is odd and v is even. The dual of this theoem can be witten as Theoem 2.9 Let k 2 and a 1,...,a k be natual numbes. The following popety holds lcm(a i1,...,a iu ) 1 i gcd(a 1,a 2,...,a k ) = 1 <...<i u k lcm(a i1,...,a iv ), (35) 1 i 1 <...<i v k whee u is odd and v is even. Sketch of poof: Assuming that the multiplicities of p in a 1,...,a k ae m 1,...,m k, the poblem of poving (34) can be educed to poving the identity below. max(m 1,...,m k ) = 1 i 1 <...<i u n whee u is odd and v is even. 3 Discussion min(m i1,...,m iu ) 1 i 1 <...<i v n min(m i1,...,m iv ), (36) The eseach oiginated in the counting poblem solved by the authos in 2. Thee, the Hoadam sequences with a fixed peiod wee enumeated using a fomula involving all the pais of numbes having the same lcm, given by fomula (1). In this pape an explicit fomula fo the numbe of intege k-tuples with the same lcm n denoted by LCM(n;k), was established. Some popeties of this aithmetic function wee exploed, being shown that LCM(n; k) is multiplicative, but not completely multiplicative. The geneal fomula plays a key ole in the enumeation of genealized peiodic Hoadam sequences, chaacteized in 3. The intege sequences LCM(n; 2), LCM(n; 3), LCM(n; 4) and LCM(n;5) wee found to be indexed in the OEIS. The function GL(d,n;k) counting the k-tuples with same gcd d and lcm n was defined. Most of the popeties wee linked to the aithmetic function L(n;k) := GL(1,n;k), whose fomula was given. Function L(n; k) was multiplicative, but not completely multiplicative. Function T(n;k), enumeating k-tuples whose min is zeo and max is n was also analyzed. Novel, o existing OEIS intege sequences wee poduced fom T(n;k) and L(n;k). Finally, two dual popeties linking the gcd and lcm of k-tuples wee pesented.

10 100 O. Bagdasa Refeences 1 T. M. APOSTOL, Intoduction to Analytic Numbe Theoy, Spinge, New Yok, O. BAGDASAR and P. J. LARCOMBE, On the numbe of complex Hoadam sequences with a fixed peiod, Fibonacci Quat., Vol. 51, 4 (2013), O. Bagdasa and P. J. Lacombe, On the chaacteization of peiodic genealized Hoadam sequences, J. Diffe. Equ. Appl. (accepted). 4 R. CRANDALL and C. POMERANCE, Pime Numbes: A Computational Pespective, Spinge, New Yok, G. H. HARDY and E. M. WRIGHT An Intoduction to the Theoy of Numbes, Oxfod Univesity Pess, Oxfod, Fifth edition, Octogon Mathematical Magazine, Collection, The On-Line Encyclopedia of Intege Sequences, OEIS Foundation Inc R. R. SITARAMACHANDRA and D. SURYANABAYANA, The numbe of pais of integes with L.C.M. x, ARCH. MATH., Vol XXI (1970), D. VĂLCAN and O. D. BAGDASAR, Genealizations of some divisibility elations in N, Ceative Math. & Inf., Vol. 18, 1 (2009), F. ZUMHING and T. ANDREESCU, A Path To Combinatoics fo Undegaduates, Bikhause, 2004.