1,1

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, ....

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.

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[n2/12]]], 3]]]];

Cf. A119028, A119646.

See A103278 for least j associated with i = A103277(n).

Sequence in context: A166911 A103657 A122504 this_sequence A147042 A018492 A059020

Adjacent sequences: A103274 A103275 A103276 this_sequence A103278 A103279 A103280

nonn

David W. Wilson (davidwwilson(AT)comcast.net), Jan 27 2005

Additional comments and examples from Joseph Biberstine (jrbibers(AT)indiana.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 27 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 29 2007