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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.
8
3, 13, 39, 118, 185, 400, 511, 1022, 1287, 2574, 4279, 8558, 11777, 24377, 23554, 46111, 99085, 165490
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, ....
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
See A103278 for least j associated with i = A103277(n).
Sequence in context: A103657 A320661 A122504 * A166897 A167910 A147042
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