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 A103277 Smallest i such that there exists j such that i = x + y + z, j = x*y*z has exactly n solutions in positive integers x <= y <= z. 7
 3, 13, 39, 118, 185, 400, 511, 1022, 1287, 2574, 4279, 8558, 11777, 24377, 23554, 46111, 99085, 165490 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Least number k such that there exists n partitions of k into 3 parts each having the same product. The greatest number k such that there exists n partitions of k into 3 parts each having the same product: 18, 102, 492, 1752, ... The number of members in each "class" of the set having n partitions into 3 parts each having the same product: 12, 54, 147, 397, .... LINKS Table of n, a(n) for n=1..18. EXAMPLE 3 = 1+1+1 & 1*1*1 = 1. 13 = 6+6+1 = 9+2+2 & 6*6*1 = 9*2*2 = 36. 39 = 20+15+4 = 24+10+5 = 25+8+6 & 20*15*4 = 24*10*5 = 25*8*6 = 1200. 118 = 54+50+14 = 63+40+15 = 70+30+18 = 72+25+21 & 54*50*14 = 63*40*15 = 70*30*18 = 72*25*21 = 37800. 185 = 90+84+11 = 110+63+12 = 126+44+15 = 132+35+18 = 135+28+22 & 90*84*11 = 110*63*12 = 126*44*15 = 132*35*18 = 135*28*22 = 83160. 400 = 196+180+24 = 245+128+27 = 252+120+28 = 270+98+32 = 280+84+36 = 288+70+42 & 196*180*24 = 245*128*27 = 252*120*28 = 270*98*32 = 280*84*36 = 288*70*42 = 846720. 511 = 260+216+35 = 280+195+36 = 315+156+40 = 325+144+42 = 336+130+45 = 360+91+60 = 364+75+72 & 260*216*35 = 280*195*36 = 315*156*40 = 325*144*42 = 336*130*45 = 360*91*60 = 364*75*72 = 1965600. 1022 = 520+432+70 = 560+390+72 = 630+312+80 = 650+288+84 = 672+260+90 = 675+256+91 = 720+182+120 = 728+150+144 & 520*432*70 = 560*390*72 = 630*312*80 = 650*288*84 = 672*260*90 = 675*256*91 = 720*182*120 = 728*150*144 = 15724800. 1287 = 600+588+99 = 648+539+100 = 720+462+105 = 770+405+112 = 825+336+126 = 840+315+132 = 880+245+162 = 882+240+165 = 891+200+196 & 600*588*99 = 648*539*100 = 720*462*105 = 770*405*112 = 825*336*126 = 840*315*132 = 880*245*162 = 882*240*165 = 891*200*196 = 34927200. From Donovan Johnson, Mar 29 2010: (Start) 2574 = 198+1176+1200 = 200+1078+1296 = 210+924+1440 = 224+810+1540 = 231+768+1575 = 252+672+1650 = 264+630+1680 = 324+490+1760 = 330+480+1764 = 392+400+1782 & 198*1176*1200 = 200*1078*1296 = 210*924*1440 = 224*810*1540 = 231*768*1575 = 252*672*1650 = 264*630*1680 = 324*490*1760 = 330*480*1764 = 392*400*1782 = 279417600. 4279 = 378+1925+1976 = 380+1820+2079 = 385+1710+2184 = 399+1540+2340 = 429+1330+2520 = 440+1274+2565 = 504+1045+2730 = 532+975+2772 = 550+936+2793 = 637+792+2850 = 684+735+2860 & 378*1925*1976 = 380*1820*2079 = 385*1710*2184 = 399*1540*2340 = 429*1330*2520 = 440*1274*2565 = 504*1045*2730 = 532*975*2772 = 550*936*2793 = 637*792*2850 = 684*735*2860 = 1437836400. 8558 = 756+3850+3952 = 760+3640+4158 = 770+3420+4368 = 798+3080+4680 = 858+2660+5040 = 880+2548+5130 = 896+2475+5187 = 1008+2090+5460 = 1064+1950+5544 = 1100+1872+5586 = 1274+1584+5700 = 1368+1470+5720 & 756*3850*3952 = 760*3640*4158 = 770*3420*4368 = 798*3080*4680 = 858*2660*5040 = 880*2548*5130 = 896*2475*5187 = 1008*2090*5460 = 1064*1950*5544 = 1100*1872*5586 = 1274*1584*5700 = 1368*1470*5720 = 11502691200. 11777 = 171+5600+6006 = 175+4914+6688 = 198+3675+7904 = 224+3003+8550 = 228+2925+8624 = 240+2717+8820 = 245+2640+8892 = 385+1512+9880 = 416+1386+9975 = 462+1235+10080 = 540+1045+10192 = 600+936+10241 = 637+880+10260 & 171*5600*6006 = 175*4914*6688 = 198*3675*7904 = 224*3003*8550 = 228*2925*8624 = 240*2717*8820 = 245*2640*8892 = 385*1512*9880 = 416*1386*9975 = 462*1235*10080 = 540*1045*10192 = 600*936*10241 = 637*880*10260 = 5751345600. 24377 = 1196+11400+11781 = 1197+11220+11960 = 1232+9690+13455 = 1254+9200+13923 = 1360+7722+15295 = 1520+6435+16422 = 1547+6270+16560 = 1748+5304+17325 = 1890+4807+17680 = 1932+4680+17765 = 2244+3933+18200 = 2261+3900+18216 = 2448+3575+18354 = 2907+2990+18480 & 1196*11400*11781 = 1197*11220*11960 = 1232*9690*13455 = 1254*9200*13923 = 1360*7722*15295 = 1520*6435*16422 = 1547*6270*16560 = 1748*5304*17325 = 1890*4807*17680 = 1932*4680*17765 = 2244*3933*18200 = 2261*3900*18216 = 2448*3575*18354 = 2907*2990*18480 = 160626866400. 23554 = 342+11200+12012 = 350+9828+13376 = 351+9728+13475 = 396+7350+15808 = 448+6006+17100 = 456+5850+17248 = 480+5434+17640 = 490+5280+17784 = 665+3584+19305 = 770+3024+19760 = 832+2772+19950 = 924+2470+20160 = 1080+2090+20384 = 1200+1872+20482 = 1274+1760+20520 & 342*11200*12012 = 350*9828*13376 = 351*9728*13475 = 396*7350*15808 = 448*6006*17100 = 456*5850*17248 = 480*5434*17640 = 490*5280*17784 = 665*3584*19305 = 770*3024*19760 = 832*2772*19950 = 924*2470*20160 = 1080*2090*20384 = 1200*1872*20482 = 1274*1760*20520 = 46010764800. (End) From Duncan Moore, Sep 02 2017: (Start) 46111 = 4446+20160+21505 = 4455+19760+21896 = 4576+17595+23940 = 4680+16560+24871 = 4725+16192+25194 = 4807+15600+25704 = 4928+14858+26325 = 5100+13984+27027 = 5187+13600+27324 = 5520+12376+28215 = 5610+12096+28405 = 5712+11799+28600 = 6270+10465+29376 = 7360+8721+30030 = 7735+8280+30096 = 7904+8100+30107 & 4446*20160*21505 = 4455*19760*21896 = 4576*17595*23940 = 4680*16560*24871 = 4725*16192*25194 = 4807*15600*25704 = 4928*14858*26325 = 5100*13984*27027 = 5187*13600*27324 = 5520*12376*28215 = 5610*12096*28405 = 5712*11799*28600 = 6270*10465*29376 = 7360*8721*30030 = 7735*8280*30096 = 7904*8100*30107 = 1927522396800. 99085 = 3770+47120+48195 = 3780+45240+50065 = 3952+37758+57375 = 3978+37107+58000 = 4176+33250+61659 = 4199+32886+62000 = 4216+32625+62244 = 4495+29070+65520 = 4500+29016+65569 = 4914+25296+68875 = 5320+22620+71145 = 7280+15390+76415 = 7395+15120+76570 = 7905+14040+77140 = 8370+13195+77520 = 9367+11718+78000 = 9945+11020+78120 & 3770*47120*48195 = 3780*45240*50065 = 3952*37758*57375 = 3978*37107*58000 = 4176*33250*61659 = 4199*32886*62000 = 4216*32625*62244 = 4495*29070*65520 = 4500*29016*65569 = 4914*25296*68875 = 5320*22620*71145 = 7280*15390*76415 = 7395*15120*76570 = 7905*14040*77140 = 8370*13195*77520 = 9367*11718*78000 = 9945*11020*78120 = 8561475468000. 165490 = 14000+72488+79002 = 14022+71500+79968 = 14080+69615+81795 = 14280+65520+85690 = 14432+63308+87750 = 14820+59040+91630 = 14896+58344+92250 = 16236+49504+99750 = 16380+48790+100320 = 16830+46740+101920 = 17290+44880+103320 = 17589+43776+104125 = 18720+40180+106590 = 19152+39000+107338 = 20090+36720+108680 = 21648+33592+110250 = 23940+30030+111520 = 25840+27720+111930 & 14000*72488*79002 = 14022*71500*79968 = 14080*69615*81795 = 14280*65520*85690 = 14432*63308*87750 = 14820*59040*91630 = 14896*58344*92250 = 16236*49504*99750 = 16380*48790*100320 = 16830*46740*101920 = 17290*44880*103320 = 17589*43776*104125 = 18720*40180*106590 = 19152*39000*107338 = 20090*36720*108680 = 21648*33592*110250 = 23940*30030*111520 = 25840*27720*111930 = 80173757664000 (End) MATHEMATICA tanya[n_] := tanya[n] = Max[Length /@ Split[ Sort[Times @@@ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers, Round[n^2/12]]], 3]]]]; CROSSREFS Cf. A060277, A119028, A119646, A317578. See A103278 for least j associated with i = A103277(n). Sequence in context: A103657 A320661 A122504 * A166897 A167910 A147042 Adjacent sequences: A103274 A103275 A103276 * A103278 A103279 A103280 KEYWORD nonn,more AUTHOR David W. Wilson, Jan 27 2005 EXTENSIONS Additional comments and examples from Joseph Biberstine (jrbibers(AT)indiana.edu) and Robert G. Wilson v, Jul 27 2006 Edited by N. J. A. Sloane, Apr 29 2007 a(10)-a(15) from Donovan Johnson, Mar 29 2010 a(16)-a(18) from Duncan Moore, Sep 02 2017 STATUS approved

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Last modified June 16 16:46 EDT 2024. Contains 373432 sequences. (Running on oeis4.)