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1 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN Geetig Futio fo G spt D. K. Humeddy #, K. Jkmm * # Deptmet of Memtis, Hidu College, Gutu, 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 Pge 73
2 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 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 Pge 74
3 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 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] 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 Pge 75
4 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN (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 we hve 2, 2 2, G3 spt , , , Coolly: The geetig futio fo, whih e multiples of is G A e umbe of smllest pts of e,, G A oveg ptitios of Coolly: The geetig futio fo, G A e umbe of smllest pts of e oveg ptitios of whih e multiples of is Pge 76
5 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 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 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. Pge 77
6 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN 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, By usig m lb e R.H.S be simplified ito 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, 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 Pge 78
7 Itetiol Joul of Ltest Tehology i Egieeig, Mgemet & Applied Siee (IJLTEMAS) Volume VI, Issue VIIIS, August 207 ISSN , 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), [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 [4]. Seg G.V.R.K: A study of M2 ptitios. Thesis submitted to Ahy Ngu Uivesity fo wd of PhD i Memtis (207) Pge 79
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