cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A321993 Numbers having less than 3 unique partitions into exactly 3 parts with the same product: complement of A119028.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 66, 67, 68, 69, 72, 73, 80, 86, 96, 102
Offset: 1

Views

Author

M. F. Hasler, Nov 23 2018

Keywords

Comments

It is known that A119028 contains all integers >= 103, see second comment in A103277.

Crossrefs

Cf. A119028, A103277.

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.

Original entry on oeis.org

3, 13, 39, 118, 185, 400, 511, 1022, 1287, 2574, 4279, 8558, 11777, 24377, 23554, 46111, 99085, 165490
Offset: 1

Views

Author

David W. Wilson, Jan 27 2005

Keywords

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, ....

Examples

			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)

Crossrefs

Cf. A060277, A119028, A119646, A317578.
See A103278 for least j associated with i = A103277(n).

Programs

  • 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]]]];

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

A119646 a(n) = maximum number of partitions of n into 3 parts, each having the same product.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3
Offset: 3

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu) and Robert G. Wilson v, Jul 27 2006

Keywords

Examples

			a(3)=1, because there is only one way to partition 3.
a(13)=2, because 13 = 6+6+1 = 9+2+2 and 6*6*1 = 9*2*2 = 36.
a(39)=3, because 39 = 20+15+4 = 24+10+5 = 25+8+6 and 20*15*4 = 24*10*5 = 25*8*6 = 1200.
See A103277 for more examples.

Crossrefs

Cf. A103277, A119028.

Programs

  • Mathematica
    pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers,(* failsafe *) Round[n^2/12]]], 3]] ]]];
Table[ tanya@n, {n, 4, 108}]
Table[SortBy[Tally[Times@@@IntegerPartitions[n,{3}]],Last][[-1,2]],{n,3,110}] (* Harvey P. Dale, Jan 08 2023 *)

Extensions

Name clarified by Dmitry Kamenetsky, Aug 02 2015

A317388 a(n) is the smallest number having at least n partitions into n parts with the same product.

Original entry on oeis.org

39, 24, 25, 26, 28, 30, 31, 34, 35, 37, 39, 41, 43, 44, 46, 48, 49, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108
Offset: 3

Views

Author

Giovanni Resta, Jul 27 2018

Keywords

Examples

			a(4) = 24 because 24 is the smallest number which admits 4 partitions in 4 parts with the same product:
24  = 12+5+4+3 = 10+8+3+3 = 10+6+6+2 = 9+8+5+2, and
720 = 12*5*4*3 = 10*8*3*3 = 10*6*6*2 = 9*8*5*2.

Links

Crossrefs

Cf. A119028.

Programs

  • Mathematica
    a[n_] := Block[{k=n}, While[Max[Last /@ Tally[Times @@@ IntegerPartitions[k, {n}]]] < n, k++]; k]; Array[a, 40, 3]

A356729 Numbers having at least 4 distinct partitions into exactly 3 parts with the same product.

Original entry on oeis.org

118, 130, 133, 135, 137, 140, 148, 149, 153, 155, 161, 167, 169, 174, 175, 182, 183, 185, 189, 190, 194, 195, 200, 202, 205, 206, 208, 209, 210, 213, 214, 215, 216, 217, 220, 221, 222, 223, 224, 225, 228, 229, 231, 234, 235, 236, 239, 240, 243, 244, 245, 247, 248, 249, 250, 251, 253, 254
Offset: 1

Views

Author

Tanya Khovanova, Sep 09 2022

Keywords

Comments

The smallest number that has at least 5 partitions is 185.

Examples

			118 is in this sequence because it has 4 partitions (14,50,54), (15,40,63), (18,30,70), and (21,25,72) with the same product 37800.

Crossrefs

Cf. A119028.

Programs

  • Mathematica
    Select[Range[3, 300], Max[Transpose[Tally[Apply[Times, IntegerPartitions[#, {3}], {1}]]][[2]]] >= 4 &]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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