OFFSET
1,3
COMMENTS
Conjecture I: The natural density of this sequence is 1.
Conjecture II: All integers > 13 are terms. The decomposition is not necessarily unique; for instance, 12! = 35^3 + 309^3 + 766^3 = 240^3 + 504^3 + 696^3.
Deshouillers, Hennecart, & Landreau conjecture (the DHL conjecture) that the sequence of numbers that are a sum of at most three cubes has density 0.0999425... (see links).
This lets us make a heuristic argument that all integers k > 13 are terms.
It was verified for k < 34. For k >= 34 we can use the fact that m is a term if m!/t^3 is the sum of three nonnegative cubes. The cubefree part of 34! is 2686295049620 (cf. A145642) and tau((34!/2686295049620)^(1/3)) = 792 (cf. A248780). 132 terms of corresponding 792 numbers are congruent to 4 or 5 mod 9, that is, there cannot be the sum of three cubes in these 132 terms by modular restriction. So we can see that if 34! isn't the sum of at most three cubes then 792 - 132 = 660 candidate numbers aren't the sum of at most three cubes.
So roughly, if the DHL conjecture holds and if that density can be used as a probability that holds independently for candidates then we have the probability that 34! is the sum of at most 3 cubes to be 1 - (1-0.0999425)^660 ~= 1 - 6.6*10^-31. For larger k this probability doesn't tend to decrease. (End)
LINKS
Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, On the density of sums of three cubes, ANTS-VII (2006), pp. 141-155.
EXAMPLE
0 and 1 are terms because 0! = 1! = 1 = 0^3 + 0^3 + 1^3.
2 is a term because 2! = 2 = 0^3 + 1^3 + 1^3.
4 is a term because 4! = 24 = 2^3 + 2^3 + 2^3.
From Chai Wah Wu, Jan 18 2016: (Start)
9! = 36^3 + 52^3 + 56^3
10! = 4^3 + 96^3 + 140^3
11! = 105^3 + 222^3 + 303^3
12! = 35^3 + 309^3 + 766^3
14! = 135^3 + 3153^3 + 3822^3
15! = 1092^3 + 2040^3 + 10908^3
16! = 7644^3 + 21192^3 + 22212^3
17! = 9984^3 + 22848^3 + 69984^3
18! = 18900^3 + 54060^3 + 184080^3
19! = 131040^3 + 331200^3 + 436320^3
20! = 87490^3 + 1034430^3 + 1098440^3
21! = 59850^3 + 2072070^3 + 3481380^3 (End)
22! = 286272^3 + 8168832^3 + 8334144^3. - Altug Alkan, Aug 08 2020
From Chai Wah Wu, Aug 09 2020: (Start)
23! = 8255520^3 + 10856160^3 + 28848960^3
24! = 8648640^3 + 9918720^3 + 85216320^3
25! = 31449600^3 + 194947200^3 + 200592000^3
26! = 133526400^3 + 232377600^3 + 729590400^3
27! = 400579200^3 + 697132800^3 + 2188771200^3
28! = 745516800^3 + 3859430400^3 + 6274195200^3
29! = 6029402400^3 + 7705152000^3 + 20136664800^3
30! = 24051081600^3 + 35394105600^3 + 59154883200^3
31! = 63842385600^3 + 74054736000^3 + 196233710400^3
32! = 19948723200^3 + 392984524800^3 + 587164032000^3
33! = 757780531200^3 + 1319649408000^3 + 1812063052800^3
34! = 2423348928000^3 + 5068495555200^3 + 5322645820800^3
35! = 221937408000^3 + 1100266675200^3 + 21780043084800^3
36! = 37944351244800^3 + 43054819315200^3 + 61932511872000^3
(End)
From Altug Alkan, Aug 15-26 2020: (Start)
37! = 24795996825600^3 + 74281492454400^3 + 237157683840000^3.
38! = 117664241587200^3 + 120627079372800^3 + 803958680448000^3.
39! = 863357752857600^3 + 953842592102400^3 + 2663078850432000^3.
40! = 2918729189376000^3 + 5087164642560000^3 + 8703942863616000^3.
41! = 7755318514944000^3 + 8120284204032000^3 + 31896357292800000^3.
42! = 89122911958080000^3 + 33781805785728000^3 + 87002517970368000^3.
43! = 122523857584128000^3 + 202407941159424000^3 + 369098064631296000^3.
44! = 259725052274688000^3 + 793899570207744000^3 + 1288734012453888000^3.
45! = 406827658382745600^3 + 1201813420282675200^3 + 4902359567603097600^3.
47! = 12321320074256793600^3 + 20307078211733913600^3 + 62859559551447859200^3.
48! = 25537325843751321600^3 + 149166695523144499200^3 + 208609080169435545600^3.
50! = 1299690649834536960000^3 + 1575788569801205760000^3 + 2896698799298304000000^3.
52! = 4714930301540659200000^3 + 30326925607072174080000^3 + 37482600824578990080000^3.
57! = 2143437030275189096448000^3 + 18952651629200785047552000^3 + 32303499916146500321280000^3. (End)
From Altug Alkan, Mar 05-13 2021: (Start)
46! = 5577191426219212800^3 + 6443840881904025600^3 + 17169667908109516800^3.
49! = 671664000771219456000^3 + 662061074870587392000^3 + 247029110344912896000^3.
51! = 9256160466097459200000^3 + 9117812465538416640000^3 + 428071307793592320000^3.
53! = 162171341319623860224000^3 + 14768160510292180992000^3 + 18786201326150049792000^3.
54! = 545218231179130629120000^3 + 335022509605704560640000^3 + 314703105438452290560000^3.
55! = 1946744272579774187520000^3 + 1230901820453108643840000^3 + 1511561473478381445120000^3.
58! = 52226010170722243215360000^3 + 102552481007618403041280000^3 + 104144718055889686855680000^3.
59! = 496516081488480416563200000^3 + 247419327579970911805440000^3 + 104213060097975874805760000^3. (49,51,53,54,55,59 found by Bernard Landreau, Mar 05-10 2021)
From Bernard Landreau, Feb 10 2023: (Start)
56! = 8440722823838300835840000^3 + 1539870961334538792960000^3 + 4732343335270526976000000^3.
60! = 1954690295686184458321920000^3 + 187526160279422365040640000^3 + 945736839075280596664320000^3.
61! = 6987261145735262954225664000^3 + 5500819928796737985183744000^3 + 3511150067368879423488000^3.
62! = 28126020674003772660940800000^3 + 12303713179773215087247360000^3 + 19449735813987841779056640000^3.
63! = 106514918440099777554186240000^3 + 49252742968526796125306880000^3 + 86830960771932156207267840000^3.
64! = 426059673760399110216744960000^3 + 197010971874107184501227520000^3 + 347323843087728624829071360000^3.
65! = 1825857768347463635450265600000^3 + 1233646969650476271309619200000^3 + 656708896142403679243468800000^3.
66! = 7629164545500731715435233280000^3 + 383304147481048793646366720000^3 + 4645292541653757960968601600000^3.
67! = 32138800724565658662277939200000^3 + 3987806882839318432102809600000^3 + 14753675466796017234670387200000^3.
68! = 121268519043338230583014195200000^3 + 74635666310379772757724364800000^3 + 65491151303650959730645401600000^3.
69! = 440198819826578009858742681600000^3 + 217119306274746004582406553600000^3 + 422815083063767403026566348800000^3.
(End)
From Bernard Landreau, Apr 12 2023: (Start)
70! = 1684880479643468059918290124800000^3 + 1267939232313822071989803417600000^3 + 1727697134569562112035900620800000^3.
71! = 7869526037543841297006565785600000^3 + 4179944826601729536159999590400000^3 + 6619802079654886665835708416000000^3.
72! = 13437726338581697013357713817600000^3 + 10167574949678977741805794099200000^3 + 38654599603517743131172247961600000^3.
73! = 96869296261623898801464382586880000^3 + 80774308520159270283270497894400000^3 + 144769602970826932947390114693120000^3.
74! = 649373800890254088606178494873600000^3 + 363407978539450964422332584755200000^3 + 207722030872866958396078844313600000^3.
75! = 2347486647113944742227212238848000000^3 + 2199783184771995658848232636416000000^3 + 1070862876804260107568106602496000000^3.
76! = 12262054139494209011130556907520000000^3 + 2762109848253646350901295382528000000^3 + 2746796636906395254645335359488000000^3.
77! = 47421174895780818749100971655168000000^3 + 18679208068237422355741320413184000000^3 + 31756770658228697228286202871808000000^3.
78! = 141193533844368458064892797124608000000^3 + 108335094312749634096990256889856000000^3 + 193437233894764827340173357613056000000^3.
79! = 897795952124597047877074078334976000000^3 + 151955762572905091739065815367680000000^3 + 551184446076431732583718393774080000000^3.
80! = 3554290394480645556188266337402880000000^3 + 1989394527958598219192394328571904000000^3 + 2658759141945971588173630544019456000000^3.
(End)
MAPLE
isA267414 := proc(n)
local nf, x, y ;
nf := n! ;
for x from 0 do
if 3*x^3 > nf then
return false;
end if;
for y from x do
if x^3+2*y^3 > nf then
break;
end if;
if isA000578(nf-x^3-y^3) then
return true;
end if;
end do:
end do:
end proc:
for n from 0 to 1000 do
if isA267414(n) then
print(n) ;
end if;
end do: # R. J. Mathar, Jan 23 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 14 2016
EXTENSIONS
a(51)-a(64) from Bernard Landreau, Feb 10 2023
a(65)-a(75) from Bernard Landreau, Apr 12 2023
STATUS
approved