A076031

[Hans Havermann]

> %I A076031
> %S A076031 1,2,8,3,4,12,5,6,7,210,9,10,11,12,330,13,14,15,16,17,46410
> %N A076031 Group the natural numbers so that the n-th group contains 
> the smallest set of n numbers whose product is a s\
>   quare: (1), (2, 8), (3, 4, 12), (5, 6, 7, 210), (9, 10, 11, 12, 
> 330), (13, 14, 15, 16, 17, 46410), ...
> %e A076031 1; 2,8; 3,4,12; 5,6,7,210; 9,10,11,12,330; 
> 13,14,15,16,17,46410; ...
> %Y A076031 Cf. A076027, A076028, A076029, A076030.
> %K A076031 more,nonn,tabl
> %O A076031 1,2
> %A A076031 Amarnath Murthy (amarnath_murthy at yahoo.com), Oct 01 2002

I'm not comfortable with the lack of specificity of the word 
"smallest". Are we constrained merely to using (in the nth grouping) 
the smallest still-available (n-1) terms? [Other, more interesting, 
constraints come to mind. Some other time, perhaps.]

At any rate, 12 appears twice. The fifth grouping should be:

{9, 10, 11, 13, 1430}

Then (I think)...

{14, 15, 16, 17, 18, 1785}
{19, 20, 21, 22, 23, 24, 168245}
{25, 26, 27, 28, 29, 30, 31, 409045}
{32, 33, 34, 35, 36, 37, 38, 39, 119629510}
{40, 41, 42, 43, 44, 45, 46, 47, 48, 293493662}

...and so on.

I've put a bunch of the hopefully correct A076031 terms at:

http://odo.ca/~haha/j/seq/a076031

Of course, all of the dependent series (A076027, A076028, A076029, 
A076030) are now in error. (In fact, A076030 introduces a new error by 
losing a digit from the fourth term.)

A076027 [currently: 1,2,3,5,9,13,18]

1, 2, 3, 5, 9, 14, 19, 25, 32, 40, 49, 59, 70, 82, 95, 109, 124, 140, 
157, 175, 194, 215, 236, 258, 281, 305, 330, 356, 383, 411, 440, 470, 
501, 533, 566, 600, 635, 671, 708, 746, 785, 825, 866, 908, 951, 995, 
1040, 1086, 1133, 1181, 1230, 1280, 1331, 1383, 1437, 1491, 1546, 1602, 
1659, 1717, 1776, 1837, 1898, 1960, 2023, 2087, 2152, 2218, 2285, 2353, 
2422, 2492, 2563, 2635, 2708, 2782, 2857, 2933, 3010, 3088, 3167, 3247, 
3328, 3410, 3493, 3577, 3662, 3748, 3835, 3923, 4012, 4102, 4193, 4285, 
4378, 4472, 4567, 4663, 4760, 4858, 4957, 5057, 5158, 5260, 5363, 5467, 
5572, 5678, 5785, 5893, 6002, 6112, 6223, 6335, 6448, 6562, 6677, 6793, 
6910, 7028, 7147, ...

A076028 [currently: 1,8,12,210,330,46410]

1, 8, 12, 210, 1430, 1785, 168245, 409045, 119629510, 293493662, 
7454211765, 8777087598279, 41162302753, 184604382197975454, 
3700467339975406, 20717472096185651930, 150297492924069082735, 
588852089711321510, 10735054115551277931604130, 
418262006588780253317218710, 13923074350280699330220534047, 
25469903971033104161820226839, 884454639879986794704109453, 
44490230948355438086564257936943, 48174687626734276280369818407465, 
168932813614006777421423281322851443, 
115719710079682124982481169178020841875130, 
2842067100782988758723814075562400794530, 
4919323311608345883443767113077990312810495, 
353600661139603431322234434625226797958994958, 
948588069298461353910785190237969964628435741970, 
644034530898513117973114365194539404725792854, ...

A076029 [currently: 1,16,144,44100,3920400,34462209600]

1, 16, 144, 44100, 18404100, 1835265600, 16304474894400, 
5421097109610000, 296757450928652313600, 178616855049415417958400, 
1411446839588456089779360000, 638977904981699070278114213990400, 
1394457904134474911759934466560000, 
3462621941531064441686873044941895024742400, 
45079514016863751764842525660275310632960000, 
189968879209747907748003335998448658758092892160000, 
1189842643063948860680204192010313841257095991296000000, 
4575163954288914114018868721942266736902736646144000000, 
92325941960445023932161684310685993082953641692029928007434240000, 
4459932152522745058813049518093490478326874061695234281854009344000000, 
...

A076030 [currently: 1,4,12,21,1980,185640]

1, 4, 12, 210, 4290, 42840, 4037880, 73628100, 17226649440, 
422630873280, 37569227295600, 25278012283043520, 37342441057521600, 
1860812172555592576320, 6714127941651376646400, 
13782919836150390515990400, 1090799084645723774857536000, 
2138963289607587938983488000, 303851842121197323393999522163200, 
66782723458412094462451023213312000, 
14449878929857400094638222349629529600, 
793008109008128685521105591935856256000, 
6049796079355507252129758551147142144000, 
58178734637564738139211045289080921377024000, 
2713062367369540557114001035839849199379968000, 
7510909749423375692817396208187762799219042816000, 
302647397487183104854726147953538967779777115335680000, 
2400857901690171686119063165345626090493116282147840000, 
5249224370036010385592707535676891541883902463960002560000, 
2424796868544112974869955566981620757238619058963934836736000, 
7085417864517718697634121530785062081378472427907126472171520000, 
10772251211961888470724678780497701655290728254833477311897600000, ...


--
"...lucky for hunters that they don't have to clean up the mess."
[Miroslav Holub, in his essay 'Shedding Life']