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Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 Geetig Futio fo G spt D. K. Humeddy #, K. Jkmm * # Deptmet of Memtis, Hidu College, Gutu, 52202 A.P, Idi * Deptmet of Memtis, 8 Mi, AECS Lyout, B, BLOCK, Sigsd, Bgloe, 5604 Idi Abstt: I is setio, we deive fomul fo e geetig futio e umbe of smllest pts iludig epetitios i ll ove G ptitio of. I. INTRODUCTION C otel d Love oy [] iitited e study of ove ptitios. Hum Reddy d Jkmm [3] deived fomul fo e umbe of i ove G ptitio of Whe e pts e i A.P. I is ppe, we obti fomul fo e geetig futio fo e umbe of smllest pts iludig epetitios i ll ove G ptitio of.. Defiitios d ottio: ) A -ptitio of is o-deesig seuee of positive iteges whose sum is =. Eh umbe is lled ptitio 2) A ove ptitio of is -ptitio of i whih pt is ove lied times t its fist ppees. 3) The dility of e set of 4), ove ptitios of is deoted by p. p s : The umbe of ove ptitios of wi lest pt gete o eul to s is defied by p s,. 5) G ptitio k : A ptitio of is G ptitio if smllest pts e of e fom, k N. 6) ove G ptitio Is G ptitio i whih fist (euivletly, e fil) ouee of pt is ove lied up to times suessively. 7) G spt : deotes e umbe of smllest pts iludig epetitios i ll 8) Ove ptitio of (, ) : 9) ( ) : (,) 0) d(, ) is e umbe of divisos of of e fom? II. GENARATING FUNCTIONS: k, k N I is setio, we obti fomul fo e geetig futio fo e umbe of smllest pts iludig epetitios i ll ove G ptitios of www.iltems.i Pge 73

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 2. Popositio: The umbe of ptitios of ito pts eh Poof: Let k k k ie. p, p (2.3(i)) wite k (, 2,..., ),,, 2,... whee.fue if, 2,...,... ley eoespodee,...,... k = umbe of ptitios of ito pts. d e k k k k i i i i is oe oe d oto fom ptitios of wi smllest pt gete k d ll ptitios of k.is oe- oe oespodee yields e euied eulity. 2.2 Coolly: Fo = e umbe of -ove ptitios of wi pts, gete o eul to k is e umbe of oveptitios of k pts. We ow deive e geetig futio fo e umbe of smllest pts of ll oveg ptitios of. oveg ptitios of wi e help of 2.3 Theoem: The geetig futio fo G spt is Poof:,, G spt Fom eoem (3.) of [3] we hve...(a) k k, k k, d, G spt p t p t k t k t k k k k Reple by, by t fo fist pt d by t fo seod pt i (2.5) G spt k k k k G spt p t p t k k t k t d, k k k t k t k t k k k t,, k k t k t k k k k k k t t,, k Fom www.iltems.i Pge 74

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 k k,, k k t t k t k t k k k, k k k k, k k k k k k k k k,, k k k k k k k k k k k k k 0 0 k k k k k k k k k k 0 0 k k k k k,,,, k k k k k, k k, k, k k k k k, k k k k,, fom [3] 2.3.4 Illusttio: We expli ou eoem by illusttio. I is otext we ofte ome oss wi tems of e fom. we eple ese tems by e powe seies expsios x k k www.iltems.i Pge 75

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 (Wi ltete The oeffiiet of x 2 x x... x... d sigs fo x d sigs fo x i e deomito. ) x i is otext be foud wi e help of M lb. Whe = 2 d = 3 i eoem 2.3.3 we hve 2, 2 2, 3 2 3 G3 spt. 3 3 3 2 2, 2 2 3 2 3 2 2 2 2 2 2 9 2 3 4 5 6 7 8 2 9 2 3 4 5 6 7 8 9 2 2 2 2 2 2 2 2 2 2 3 2 2, 3 2 3 2 2 2 2 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 2 2 2 2 2 2 2 2, 4 8 4 25 58 8 226 448... 2 3 4 5 6 7 8 9 0 2 3 4 5 6 3 3 6 5 27 5 93... 2 3 4 5 6 7 4 8 4 25 58... 2 3 4 5 6 7 3 2 7 3 26 53 97... 2 3 4 5 6 7 3 6 2 39 78 59 29......... 2.3.5 Coolly: The geetig futio fo, whih e multiples of is G A e umbe of smllest pts of e,, G A oveg ptitios of 2.3.6 Coolly: The geetig futio fo, G A e umbe of smllest pts of e oveg ptitios of whih e multiples of is www.iltems.i Pge 76

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 G A, 2, To evlute e sum of smllest pts of oveg ptitios of, we popose e followig eoem. oveg ptitios of by pplyig e oept of 2.3.7 Theoem: The geetig futio fo e sum of smllest pts of sumg spt,, t Poof: The sum of smllest pts of oveg ptitios of is oveg ptitios of positive itege is k k k k k, k, sum G spt p t p t t k t k d, k d, k k k k k,, p t p t t k t k k k k Fist eple by, e eple by t i (2..) k k k p t t kt.. p t t kt d, k k k Hee Fom (2.2(i)) e geetig futio fo e sum of smllest pts of e k k k, k k k k k k k k k k k, k k k, k k k k k, k k oveg ptitios of. www.iltems.i Pge 77

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 k k k k k k k k k k k k 0 0,, sumg spt,, 2.8 Illusttio: Whe = 2, = 2 by 2.7 we hve sumg2spt 2, 2 2, 2 2 2 2 By usig m lb e R.H.S be simplified ito 2 3 4 5 6 7 8 9 0 3 2 8 63 96 89 336 645 053 725... The sum of smllest pts of seodoveg2 5, 5, 5, 5, 5, 5, 5, 5, 5, 4 2, 4 2, 4 2, 4 2, 4 2, 4 2, 4 2, 4 2, 4 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2 3 2, 3 2, 3 2, 3 2, 3 2, 2 2 2, 2 2 2, 2 2 2, 3, 3 3, 3, 3, 3, 3, 3, 3, 2 2, 2 2, 2 2, (fom 3 ) ptitios of 6 is 225 d is veified fom e followig www.iltems.i Pge 78

Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 2278-2540 2 2, 2 2, 2 2, 2 2, 2 2, 2 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,,,. I e bove tble, is ude lied i eh ptitio to speify its lest popety. The ove ptitios hve to be outed tkig ito osidetio e ove lies o e pts. The umbe of suh ove ptitio is 225d eefoe e sum of e lest pts (eh beig ) is 225. Akowledgemet: The uos e kful to (Rtd) pofesso I.Rmbhd sm fo his vluble suggestios d ommets duig peptio of is ppe. REFERENCES []. Sylve Coteel d Jeemy Love oy: Ove ptitios, Ts.Ame.M.So 356(4), 623-635. [2]. Humeddy.K (200). A study of -ptitios, esis submitted to Ahy Ngu uivesity fo wd of Ph.D i Memtis [3]. Hum Reddy.K, K. Jkmm "Geetig Futio fo Ptitios wi Pts i A.P" Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee-IJLTEMAS vol.6 issue 8, pp.34-39 207 [4]. Seg G.V.R.K: A study of M2 ptitios. Thesis submitted to Ahy Ngu Uivesity fo wd of PhD i Memtis (207) www.iltems.i Pge 79

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