%I #173 Apr 30 2023 18:04:16
%S 0,1,2,4,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
%U 53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80
%N Integers k such that there exist nonnegative integers x,y,z with k! = x^3 + y^3 + z^3.
%C From _Altug Alkan_, _David A. Corneth_ and _Chai Wah Wu_, Aug 09-26 2020: (Start)
%C Conjecture I: The natural density of this sequence is 1.
%C 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.
%C 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).
%C This lets us make a heuristic argument that all integers k > 13 are terms.
%C 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.
%C 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)
%H Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, <a href="https://doi.org/10.1007/11792086_11">On the density of sums of three cubes</a>, ANTS-VII (2006), pp. 141-155.
%e 0 and 1 are terms because 0! = 1! = 1 = 0^3 + 0^3 + 1^3.
%e 2 is a term because 2! = 2 = 0^3 + 1^3 + 1^3.
%e 4 is a term because 4! = 24 = 2^3 + 2^3 + 2^3.
%e From _Chai Wah Wu_, Jan 18 2016: (Start)
%e 9! = 36^3 + 52^3 + 56^3
%e 10! = 4^3 + 96^3 + 140^3
%e 11! = 105^3 + 222^3 + 303^3
%e 12! = 35^3 + 309^3 + 766^3
%e 14! = 135^3 + 3153^3 + 3822^3
%e 15! = 1092^3 + 2040^3 + 10908^3
%e 16! = 7644^3 + 21192^3 + 22212^3
%e 17! = 9984^3 + 22848^3 + 69984^3
%e 18! = 18900^3 + 54060^3 + 184080^3
%e 19! = 131040^3 + 331200^3 + 436320^3
%e 20! = 87490^3 + 1034430^3 + 1098440^3
%e 21! = 59850^3 + 2072070^3 + 3481380^3 (End)
%e 22! = 286272^3 + 8168832^3 + 8334144^3. - _Altug Alkan_, Aug 08 2020
%e From _Chai Wah Wu_, Aug 09 2020: (Start)
%e 23! = 8255520^3 + 10856160^3 + 28848960^3
%e 24! = 8648640^3 + 9918720^3 + 85216320^3
%e 25! = 31449600^3 + 194947200^3 + 200592000^3
%e 26! = 133526400^3 + 232377600^3 + 729590400^3
%e 27! = 400579200^3 + 697132800^3 + 2188771200^3
%e 28! = 745516800^3 + 3859430400^3 + 6274195200^3
%e 29! = 6029402400^3 + 7705152000^3 + 20136664800^3
%e 30! = 24051081600^3 + 35394105600^3 + 59154883200^3
%e 31! = 63842385600^3 + 74054736000^3 + 196233710400^3
%e 32! = 19948723200^3 + 392984524800^3 + 587164032000^3
%e 33! = 757780531200^3 + 1319649408000^3 + 1812063052800^3
%e 34! = 2423348928000^3 + 5068495555200^3 + 5322645820800^3
%e 35! = 221937408000^3 + 1100266675200^3 + 21780043084800^3
%e 36! = 37944351244800^3 + 43054819315200^3 + 61932511872000^3
%e (End)
%e From _Altug Alkan_, Aug 15-26 2020: (Start)
%e 37! = 24795996825600^3 + 74281492454400^3 + 237157683840000^3.
%e 38! = 117664241587200^3 + 120627079372800^3 + 803958680448000^3.
%e 39! = 863357752857600^3 + 953842592102400^3 + 2663078850432000^3.
%e 40! = 2918729189376000^3 + 5087164642560000^3 + 8703942863616000^3.
%e 41! = 7755318514944000^3 + 8120284204032000^3 + 31896357292800000^3.
%e 42! = 89122911958080000^3 + 33781805785728000^3 + 87002517970368000^3.
%e 43! = 122523857584128000^3 + 202407941159424000^3 + 369098064631296000^3.
%e 44! = 259725052274688000^3 + 793899570207744000^3 + 1288734012453888000^3.
%e 45! = 406827658382745600^3 + 1201813420282675200^3 + 4902359567603097600^3.
%e 47! = 12321320074256793600^3 + 20307078211733913600^3 + 62859559551447859200^3.
%e 48! = 25537325843751321600^3 + 149166695523144499200^3 + 208609080169435545600^3.
%e 50! = 1299690649834536960000^3 + 1575788569801205760000^3 + 2896698799298304000000^3.
%e 52! = 4714930301540659200000^3 + 30326925607072174080000^3 + 37482600824578990080000^3.
%e 57! = 2143437030275189096448000^3 + 18952651629200785047552000^3 + 32303499916146500321280000^3. (End)
%e From _Altug Alkan_, Mar 05-13 2021: (Start)
%e 46! = 5577191426219212800^3 + 6443840881904025600^3 + 17169667908109516800^3.
%e 49! = 671664000771219456000^3 + 662061074870587392000^3 + 247029110344912896000^3.
%e 51! = 9256160466097459200000^3 + 9117812465538416640000^3 + 428071307793592320000^3.
%e 53! = 162171341319623860224000^3 + 14768160510292180992000^3 + 18786201326150049792000^3.
%e 54! = 545218231179130629120000^3 + 335022509605704560640000^3 + 314703105438452290560000^3.
%e 55! = 1946744272579774187520000^3 + 1230901820453108643840000^3 + 1511561473478381445120000^3.
%e 58! = 52226010170722243215360000^3 + 102552481007618403041280000^3 + 104144718055889686855680000^3.
%e 59! = 496516081488480416563200000^3 + 247419327579970911805440000^3 + 104213060097975874805760000^3. (49,51,53,54,55,59 found by _Bernard Landreau_, Mar 05-10 2021)
%e From _Bernard Landreau_, Feb 10 2023: (Start)
%e 56! = 8440722823838300835840000^3 + 1539870961334538792960000^3 + 4732343335270526976000000^3.
%e 60! = 1954690295686184458321920000^3 + 187526160279422365040640000^3 + 945736839075280596664320000^3.
%e 61! = 6987261145735262954225664000^3 + 5500819928796737985183744000^3 + 3511150067368879423488000^3.
%e 62! = 28126020674003772660940800000^3 + 12303713179773215087247360000^3 + 19449735813987841779056640000^3.
%e 63! = 106514918440099777554186240000^3 + 49252742968526796125306880000^3 + 86830960771932156207267840000^3.
%e 64! = 426059673760399110216744960000^3 + 197010971874107184501227520000^3 + 347323843087728624829071360000^3.
%e 65! = 1825857768347463635450265600000^3 + 1233646969650476271309619200000^3 + 656708896142403679243468800000^3.
%e 66! = 7629164545500731715435233280000^3 + 383304147481048793646366720000^3 + 4645292541653757960968601600000^3.
%e 67! = 32138800724565658662277939200000^3 + 3987806882839318432102809600000^3 + 14753675466796017234670387200000^3.
%e 68! = 121268519043338230583014195200000^3 + 74635666310379772757724364800000^3 + 65491151303650959730645401600000^3.
%e 69! = 440198819826578009858742681600000^3 + 217119306274746004582406553600000^3 + 422815083063767403026566348800000^3.
%e (End)
%e From _Bernard Landreau_, Apr 12 2023: (Start)
%e 70! = 1684880479643468059918290124800000^3 + 1267939232313822071989803417600000^3 + 1727697134569562112035900620800000^3.
%e 71! = 7869526037543841297006565785600000^3 + 4179944826601729536159999590400000^3 + 6619802079654886665835708416000000^3.
%e 72! = 13437726338581697013357713817600000^3 + 10167574949678977741805794099200000^3 + 38654599603517743131172247961600000^3.
%e 73! = 96869296261623898801464382586880000^3 + 80774308520159270283270497894400000^3 + 144769602970826932947390114693120000^3.
%e 74! = 649373800890254088606178494873600000^3 + 363407978539450964422332584755200000^3 + 207722030872866958396078844313600000^3.
%e 75! = 2347486647113944742227212238848000000^3 + 2199783184771995658848232636416000000^3 + 1070862876804260107568106602496000000^3.
%e 76! = 12262054139494209011130556907520000000^3 + 2762109848253646350901295382528000000^3 + 2746796636906395254645335359488000000^3.
%e 77! = 47421174895780818749100971655168000000^3 + 18679208068237422355741320413184000000^3 + 31756770658228697228286202871808000000^3.
%e 78! = 141193533844368458064892797124608000000^3 + 108335094312749634096990256889856000000^3 + 193437233894764827340173357613056000000^3.
%e 79! = 897795952124597047877074078334976000000^3 + 151955762572905091739065815367680000000^3 + 551184446076431732583718393774080000000^3.
%e 80! = 3554290394480645556188266337402880000000^3 + 1989394527958598219192394328571904000000^3 + 2658759141945971588173630544019456000000^3.
%e (End)
%p isA267414 := proc(n)
%p local nf,x,y ;
%p nf := n! ;
%p for x from 0 do
%p if 3*x^3 > nf then
%p return false;
%p end if;
%p for y from x do
%p if x^3+2*y^3 > nf then
%p break;
%p end if;
%p if isA000578(nf-x^3-y^3) then
%p return true;
%p end if;
%p end do:
%p end do:
%p end proc:
%p for n from 0 to 1000 do
%p if isA267414(n) then
%p print(n) ;
%p end if;
%p end do: # _R. J. Mathar_, Jan 23 2016
%Y Cf. A000142, A000578, A003072, A003325, A004825, A226955.
%K nonn
%O 1,3
%A _Altug Alkan_, Jan 14 2016
%E a(51)-a(64) from _Bernard Landreau_, Feb 10 2023
%E a(65)-a(75) from _Bernard Landreau_, Apr 12 2023