On GCD-morphic sequences

On GCD-morphic sequences M. Dziemia´ nczuk(*), Wies la w Ba jguz(**) Institute of Computer Science, Bia lystok University PL-15- 887 Bia lystok, ul.Sosnow a 6 4, PO LAND e-mail: (*) Maciek.ciupa@ gmail.com, (**) Ba jguz@wp.pl Abstract This note is a res po nse to one of the problems p osed by Kw a ´ sniewski in [1, 2], see als o [3] i.e. GCD-morphic Pr oblem I II. W e show that any GCD-morphic sequence F is a t the po in t pro duct of primary GCD- morphic sequences and any GCD-morphic seq uence is enco ded by nat- ural num ber v alued seq uence satisfying condition (C1). The pr oblem of genera l imp ortance - for ex ample in n umber theory was formulated in [1 , 2] while in vestigating a new class o f DA G’s and their corr esp on- dent p.o. sets enco de d uniquely by sequences with combinatorially int erpr etable prop erties. Key W o rd s: GCD-morphic sequence AMS Classification Num b ers: 06A07, 05A10, 11A41, 05C20 Presen ted at Gian-Carlo Po lish Seminar: http://ii.uwb.e du.pl/akk/sem/sem r ota.htm 1 Preliminaries Definition 1 (GCD-morphic sequence [1]) The inte gers’ value d se quenc e F = { F n } n ≥ 0 is c al le d the GCD-morphic se quenc e if GC D ( F n , F m ) = F GC D ( n,m ) wher e GC D stays for Gr e atest Common Divisor op er ator. GCD-morphism Problem. Problem I I I. [1] Find effe ctive char ac- terizations and/or an algorithm to pr o duc e the GCD-morphic se quenc es i.e. find al l examples. Definition 2 A ny natur al numb ers value d se quenc e G of the form G c,N ≡ { g n } n ≥ 0 , wher e g n =  c N | n 1 n = 0 ∨ N ∤ n is c al le d the primary GCD-morphic se quenc e. Observ ation 1 F or any natur al c > 0 , G c,k is GCD-morphic. 1 Notation The pro duct of t w o sequences F n = ( F n 1 , F n 2 , . . . ), F k = ( F k 1 , F k 2 , . . . ) is at the p oin t pr o duct of functions i.e. F n · F k = ( F n 1 · F k 1 , F n 2 · F k 2 , . . . ). 2 GCD-morphic sequences Lemma 1 L et F n = (1 , . . . , 1 , F n n , F n n +1 , . . . ) b e a GCD-morphic se quenc e. Then we have the e quality F n = G F n n ,n · F n +1 , wher e F n +1 = { F n +1 k } ∞ k =1 , F n +1 k =    1 , if k ≤ n ; F n k F n n , if n | k ; F n k , otherwise. and F n +1 is GCD- morphic. Pro of. Since F n is GCD-morphic, it fo llo ws that F n n | F n k for n | k . Therefore F n +1 k is w ell defin ed and the equalit y is obvious. No w we must pr o of that F n +1 is GCD-morphic. Let k , l b e any natural num b ers. W e m ust sho w, that C GD -morp hic condi- tion: GC D ( F n +1 k , F n +1 l ) = F n +1 GC D ( k ,l ) is tru e. I n order th is to b e prov ed let us in ve stigate the follo wing cases: a) n 6 | k and n 6 | l , b) n | k and n | l , c) otherwise. In th e a) case F n +1 k = F n k , F n +1 l = F n l and F n +1 GC D ( k ,l ) = F n GC D ( k ,l ) holds. Since F n is GCD-morp hic, therefore the C GD -morphic condition is satified. In the case b) w e obtain n | GC D ( k , l ) and accordingly F n +1 k = F n k F n n , F n +1 l = F n l F n n and F n +1 GC D ( k ,l ) = F n GC D ( k,l ) F n n . It pro vides to GC D ( F n +1 k , F n +1 l ) = GC D ( F n k ,F n l ) F n n = F n GC D ( k,l ) F n n = F n +1 GC D ( k ,l ) T o pro ve that C GD -morph ic condition is satified in the case c) let us assume that n | k and n 6 | l . Then F n k = F n +1 k · F n n , F n +1 l = F n l and F n +1 GC D ( k ,l ) = F n GC D ( k ,l ) and consequen tly GC D ( F n +1 k · F n n , F n +1 l ) = GC D ( F n k , F n l ) = F n GC D ( k ,l ) = F n +1 GC D ( k ,l ) . A contrario p ro of: assu me that GC D ( F n +1 k · F n n , F n +1 l ) 6 = GC D ( F n +1 k , F n +1 l ). It is equiv alent to GC D ( F n +1 k · F n n , F n l ) 6 = GC D ( F n +1 k , F n l ). Then GC D ( F n n , F n l ) 6 = 1, but otherwise GC D ( n, l ) < n and consequen tly GC D ( F n n , F n l ) = F n GC D ( n,l ) = 1. Hence - cont radiction. Therefore GC D ( F n +1 k , F n +1 l ) = GC D ( F n +1 k · F n n , F n +1 l ) = F n +1 GC D ( k ,l ) .  2 Since every p eriod ic sequence ((1 , . . . , 1 , F n n )) is GCD-morph ic then ac- cording to the ab o v e lemma w e sum up. Lemma 1 provides a prescription for ho w t o pro d uce infinite num b er of primary GCD-morphic sequences. Re- call: ev ery su c h pro duct sequence is well defined i.e. eac h elemen t of this sequence is a pro du ct of finite num b er elemen ts different from 1. Conclusion Let F = ( F 1 1 , F 1 2 , . . . ) b e a GCD-morphic sequence. Then F is a pro du ct G c 1 , 1 · G c 2 , 2 · . . . of primary GCD-morph ic sequen ces, where c n = F 1 n Q k | n,k 1. Then GC D ( F n , F k ) = F α . Since GC D ( n, k ) = α 6 = k , then α | k ∧ α < k , and therefore (1) F k = Q j | k c j = Q j | α · c k · F ′ k . Moreo v er f rom GC D ( n, k ) = α 6 = k w e ha ve α | n ∧ α < n and it leads to (2) F n = Q j | n c j = Q j | α · c n · F ′ n . F rom (1) and (2) we h a v e th at GC D ( F n , F k ) = Q j | α c j · GC D ( c k · F ′ k , c n · F ′ n ) ≥ F α · GC D ( c k , c n ) By (*) GC D ( c n , c k ) > 1, and therefo e GC D ( F n , F k ) > F α . Hence con tra- diction. Next, we need to sho w that if F is a pro d uct of infin ite num b er primary GCD-morphic sequences G c i ,i whic h fulfi ls (C1) then F is GCD-morphic. Since F is a pr o duct of G c i ,i , then F n = Q j | n c j for an y n . Therefore GC D ( F n , F k ) = GC D ( Q j | n c j , Q j | k c j ) = = GC D ( Q j | n ∧ j | k c j · Q j | n ∧ j ∤ k c j , Q j | n ∧ j | k c j · Q j | k ∧ j ∤ n c j ) = Q j | n ∧ j | k c j · P = = Q j | GC D ( n,k ) c j · P = F GC D ( n,k ) · P where P = GC D ( Q j | n ∧ j ∤ k c j , Q j | k ∧ j ∤ n c j ). If P > 1 then there exists r,s suc h that (*) r | n, r ∤ k , s | k , s ∤ n and GC D ( c r , c s ) > 1. 3 By (*) holds r ∤ s and s ∤ r . Then GC D ( c r , c s ) = 1 b y (C1) - con trary to (*). Hence P = 1 and GC D ( F n , F k ) = F GC D ( n,k ) . It ends the pro of  The lemmas 1 an d 2 lead to the follo wing conclusion. Conclusion An y sequence C ≡ { c n } n ≥ 1 suc h that { G c n ,n } n ≥ 1 satisfies condition ( C 1) enco des GCD-morphic sequence and otherwise - an y GCD-morphic sequence is encod ed b y su c h sequence. Th e corresp ondence is biunivoque. Examples: Natural n umb ers’ sequence C = (1 , 2 , 3 , 2 , 5 , 1 , 7 , 2 , 3 , 1 , 11 , 1 , 13 , 1 , 1 , 2 , 17 , ... ) Fib onacci n umb ers’ sequen ce C = (1 , 1 , 2 , 3 , 5 , 4 , 13 , 7 , 17 , 11 , 8 9 , 6 , ... ) The sequence of pro ducts primary divisors of natural num b ers and it is a GCD-morphic sequence enco ded b y c n = n n if n primary; 1 otherwise. . Ac kno wledgemen ts W e w ould lik e to th ank Professor A. Kr zysztof Kwa ´ sniewski for guidance and final impr o v ement s of this pap er. A ttandence of Ewa Krot-Sienia wsk a is highly appreciated to o. References [1] A. Krzysztof Kw a ´ sniewski, Cobweb p osets as nonc ommutative pr efabs , Adv. Stud. Con temp. Math. v ol. 14 (1) (200 7) 37-47. [2] A. K rzysztof K w a ´ sniewski, On c obweb p osets and their c ombinatorial ly admissible se q uenc es , arXiv:math.CO/0512578, 21 O ct 2007 . [3] A. Kr zysztof Kwa ´ sniewski, M. Dziemia´ nczuk Cobweb p osets - R e- c ent R esults , ISRAMA 2007 , Decem b er 1-17 20 07 Kolcata, INDIA, arXiv:0801 .3985 (25 Jan 2008). 4