cp's OEIS Frontend

A267414 Integers k such that there exist nonnegative integers x,y,z with k! = x^3 + y^3 + z^3.

%I A267414 #177 Aug 17 2025 03:51:29
%S A267414 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 A267414 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
%U A267414 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 A267414 Integers k such that there exist nonnegative integers x,y,z with k! = x^3 + y^3 + z^3.
%C A267414 From _Altug Alkan_, _David A. Corneth_ and _Chai Wah Wu_, Aug 09-26 2020: (Start)
%C A267414 Conjecture I: The natural density of this sequence is 1.
%C A267414 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 A267414 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 A267414 This lets us make a heuristic argument that all integers k > 13 are terms.
%C A267414 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 A267414 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 A267414 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 A267414 0 and 1 are terms because 0! = 1! = 1 = 0^3 + 0^3 + 1^3.
%e A267414 2 is a term because 2! = 2 = 0^3 + 1^3 + 1^3.
%e A267414 4 is a term because 4! = 24 = 2^3 + 2^3 + 2^3.
%e A267414 From _Chai Wah Wu_, Jan 18 2016: (Start)
%e A267414 9! = 36^3 + 52^3 + 56^3
%e A267414 10! = 4^3 + 96^3 + 140^3
%e A267414 11! = 105^3 + 222^3 + 303^3
%e A267414 12! = 35^3 + 309^3 + 766^3
%e A267414 14! = 135^3 + 3153^3 + 3822^3
%e A267414 15! = 1092^3 + 2040^3 + 10908^3
%e A267414 16! = 7644^3 + 21192^3 + 22212^3
%e A267414 17! = 9984^3 + 22848^3 + 69984^3
%e A267414 18! = 18900^3 + 54060^3 + 184080^3
%e A267414 19! = 131040^3 + 331200^3 + 436320^3
%e A267414 20! = 87490^3 + 1034430^3 + 1098440^3
%e A267414 21! = 59850^3 + 2072070^3 + 3481380^3 (End)
%e A267414 22! = 286272^3 + 8168832^3 + 8334144^3. - _Altug Alkan_, Aug 08 2020
%e A267414 From _Chai Wah Wu_, Aug 09 2020: (Start)
%e A267414 23! = 8255520^3 + 10856160^3 + 28848960^3
%e A267414 24! = 8648640^3 + 9918720^3 + 85216320^3
%e A267414 25! = 31449600^3 + 194947200^3 + 200592000^3
%e A267414 26! = 133526400^3 + 232377600^3 + 729590400^3
%e A267414 27! = 400579200^3 + 697132800^3 + 2188771200^3
%e A267414 28! = 745516800^3 + 3859430400^3 + 6274195200^3
%e A267414 29! = 6029402400^3 + 7705152000^3 + 20136664800^3
%e A267414 30! = 24051081600^3 + 35394105600^3 + 59154883200^3
%e A267414 31! = 63842385600^3 + 74054736000^3 + 196233710400^3
%e A267414 32! = 19948723200^3 + 392984524800^3 + 587164032000^3
%e A267414 33! = 757780531200^3 + 1319649408000^3 + 1812063052800^3
%e A267414 34! = 2423348928000^3 + 5068495555200^3 + 5322645820800^3
%e A267414 35! = 221937408000^3 + 1100266675200^3 + 21780043084800^3
%e A267414 36! = 37944351244800^3 + 43054819315200^3 + 61932511872000^3 (End)
%e A267414 From _Altug Alkan_, Aug 15-26 2020: (Start)
%e A267414 37! = 24795996825600^3 + 74281492454400^3 + 237157683840000^3.
%e A267414 38! = 117664241587200^3 + 120627079372800^3 + 803958680448000^3.
%e A267414 39! = 863357752857600^3 + 953842592102400^3 + 2663078850432000^3.
%e A267414 40! = 2918729189376000^3 + 5087164642560000^3 + 8703942863616000^3.
%e A267414 41! = 7755318514944000^3 + 8120284204032000^3 + 31896357292800000^3.
%e A267414 42! = 89122911958080000^3 + 33781805785728000^3 + 87002517970368000^3.
%e A267414 43! = 122523857584128000^3 + 202407941159424000^3 + 369098064631296000^3.
%e A267414 44! = 259725052274688000^3 + 793899570207744000^3 + 1288734012453888000^3.
%e A267414 45! = 406827658382745600^3 + 1201813420282675200^3 + 4902359567603097600^3.
%e A267414 47! = 12321320074256793600^3 + 20307078211733913600^3 + 62859559551447859200^3.
%e A267414 48! = 25537325843751321600^3 + 149166695523144499200^3 + 208609080169435545600^3.
%e A267414 50! = 1299690649834536960000^3 + 1575788569801205760000^3 + 2896698799298304000000^3.
%e A267414 52! = 4714930301540659200000^3 + 30326925607072174080000^3 + 37482600824578990080000^3.
%e A267414 57! = 2143437030275189096448000^3 + 18952651629200785047552000^3 + 32303499916146500321280000^3. (End)
%e A267414 From _Altug Alkan_, Mar 05-13 2021: (Start)
%e A267414 46! = 5577191426219212800^3 + 6443840881904025600^3 + 17169667908109516800^3.
%e A267414 49! = 671664000771219456000^3 + 662061074870587392000^3 + 247029110344912896000^3.
%e A267414 51! = 9256160466097459200000^3 + 9117812465538416640000^3 + 428071307793592320000^3.
%e A267414 53! = 162171341319623860224000^3 + 14768160510292180992000^3 + 18786201326150049792000^3.
%e A267414 54! = 545218231179130629120000^3 + 335022509605704560640000^3 + 314703105438452290560000^3.
%e A267414 55! = 1946744272579774187520000^3 + 1230901820453108643840000^3 + 1511561473478381445120000^3.
%e A267414 58! = 52226010170722243215360000^3 + 102552481007618403041280000^3 + 104144718055889686855680000^3.
%e A267414 59! = 496516081488480416563200000^3 + 247419327579970911805440000^3 + 104213060097975874805760000^3. (49,51,53,54,55,59 found by _Bernard Landreau_, Mar 05-10 2021) (End)
%e A267414 From _Bernard Landreau_, Feb 10 2023: (Start)
%e A267414 56! = 8440722823838300835840000^3 + 1539870961334538792960000^3 + 4732343335270526976000000^3.
%e A267414 60! = 1954690295686184458321920000^3 + 187526160279422365040640000^3 + 945736839075280596664320000^3.
%e A267414 61! = 6987261145735262954225664000^3 + 5500819928796737985183744000^3 + 3511150067368879423488000^3.
%e A267414 62! = 28126020674003772660940800000^3 + 12303713179773215087247360000^3 + 19449735813987841779056640000^3.
%e A267414 63! = 106514918440099777554186240000^3 + 49252742968526796125306880000^3 + 86830960771932156207267840000^3.
%e A267414 64! = 426059673760399110216744960000^3 + 197010971874107184501227520000^3 + 347323843087728624829071360000^3.
%e A267414 65! = 1825857768347463635450265600000^3 + 1233646969650476271309619200000^3 + 656708896142403679243468800000^3.
%e A267414 66! = 7629164545500731715435233280000^3 + 383304147481048793646366720000^3 + 4645292541653757960968601600000^3.
%e A267414 67! = 32138800724565658662277939200000^3 + 3987806882839318432102809600000^3 + 14753675466796017234670387200000^3.
%e A267414 68! = 121268519043338230583014195200000^3 + 74635666310379772757724364800000^3 + 65491151303650959730645401600000^3.
%e A267414 69! = 440198819826578009858742681600000^3 + 217119306274746004582406553600000^3 + 422815083063767403026566348800000^3. (End)
%e A267414 From _Bernard Landreau_, Apr 12 2023: (Start)
%e A267414 70! = 1684880479643468059918290124800000^3 + 1267939232313822071989803417600000^3 + 1727697134569562112035900620800000^3.
%e A267414 71! = 7869526037543841297006565785600000^3 + 4179944826601729536159999590400000^3 + 6619802079654886665835708416000000^3.
%e A267414 72! = 13437726338581697013357713817600000^3 + 10167574949678977741805794099200000^3 + 38654599603517743131172247961600000^3.
%e A267414 73! = 96869296261623898801464382586880000^3 + 80774308520159270283270497894400000^3 + 144769602970826932947390114693120000^3.
%e A267414 74! = 649373800890254088606178494873600000^3 + 363407978539450964422332584755200000^3 + 207722030872866958396078844313600000^3.
%e A267414 75! = 2347486647113944742227212238848000000^3 + 2199783184771995658848232636416000000^3 + 1070862876804260107568106602496000000^3.
%e A267414 76! = 12262054139494209011130556907520000000^3 + 2762109848253646350901295382528000000^3 + 2746796636906395254645335359488000000^3.
%e A267414 77! = 47421174895780818749100971655168000000^3 + 18679208068237422355741320413184000000^3 + 31756770658228697228286202871808000000^3.
%e A267414 78! = 141193533844368458064892797124608000000^3 + 108335094312749634096990256889856000000^3 + 193437233894764827340173357613056000000^3.
%e A267414 79! = 897795952124597047877074078334976000000^3 + 151955762572905091739065815367680000000^3 + 551184446076431732583718393774080000000^3.
%e A267414 80! = 3554290394480645556188266337402880000000^3 + 1989394527958598219192394328571904000000^3 + 2658759141945971588173630544019456000000^3. (End)
%p A267414 isA267414 := proc(n)
%p A267414     local nf,x,y ;
%p A267414     nf := n! ;
%p A267414     for x from 0 do
%p A267414         if 3*x^3 > nf then
%p A267414             return false;
%p A267414         end if;
%p A267414         for y from x do
%p A267414             if x^3+2*y^3 > nf then
%p A267414                 break;
%p A267414             end if;
%p A267414             if isA000578(nf-x^3-y^3) then
%p A267414                 return true;
%p A267414             end if;
%p A267414         end do:
%p A267414     end do:
%p A267414 end proc:
%p A267414 for n from 0 to 1000 do
%p A267414     if isA267414(n) then
%p A267414         print(n) ;
%p A267414     end if;
%p A267414 end do: # _R. J. Mathar_, Jan 23 2016
%Y A267414 Cf. A000142, A000578, A003072, A003325, A004825, A226955.
%K A267414 nonn
%O A267414 1,3
%A A267414 _Altug Alkan_, Jan 14 2016
%E A267414 a(51)-a(64) from _Bernard Landreau_, Feb 10 2023
%E A267414 a(65)-a(75) from _Bernard Landreau_, Apr 12 2023