A145642 -id:A145642 - cp's OEIS frontend

A248762 Greatest cube that divides n!.

1, 1, 1, 8, 8, 8, 8, 64, 1728, 1728, 1728, 13824, 13824, 13824, 46656000, 2985984000, 2985984000, 2985984000, 2985984000, 23887872000, 221225582592000, 221225582592000, 221225582592000, 1769804660736000, 221225582592000000, 221225582592000000

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000142, A008834, A145642, A248763.

z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 3; Table[p[m, n], {n, 1, z}]  (* A248762 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248763 *)
Table[n!/p[m, n], {n, 1, z}]      (* A145642 *)
gk[n_]:=Select[Divisors[n!],IntegerQ[Surd[#,3]]&]; Max[#]&/@Array[gk,30] (* Harvey P. Dale, Sep 16 2021 *)
f[p_, e_] := p^(3*Floor[e/3]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)

a(n)=k=ceil((n!/2)^(1/3));while(n!%k^3,k--);k^3
vector(20,n,a(n)) \\ Derek Orr, Oct 19 2014

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(3*(f[i, 2]\3)));} \\ Amiram Eldar, Sep 01 2024

a(n) = n!/A145642(n).
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A008834(n!).
a(n) = A248763(n)^3. (End)

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

Original entry on oeis.org

0, 1, 2, 4, 9, 10, 11, 12, 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, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 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

Offset: 1

Altug Alkan, Jan 14 2016

nonn

From Altug Alkan, David A. Corneth and Chai Wah Wu, Aug 09-26 2020: (Start)
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)

			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) (End)
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)

Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, On the density of sums of three cubes, ANTS-VII (2006), pp. 141-155.

Cf. A000142, A000578, A003072, A003325, A004825, A226955.

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

a(51)-a(64) from Bernard Landreau, Feb 10 2023

a(65)-a(75) from Bernard Landreau, Apr 12 2023

A248763 Greatest k such that k^3 divides n!

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 24, 24, 24, 360, 1440, 1440, 1440, 1440, 2880, 60480, 60480, 60480, 120960, 604800, 604800, 1814400, 3628800, 3628800, 3628800, 3628800, 14515200, 479001600, 479001600, 479001600, 958003200, 958003200, 958003200

Offset: 1

Clark Kimberling, Oct 14 2014

Every term divides all its successors.

			a(4) = 2 because 2^3 divides 24 and if k > 2, then k^3 > 8 does not divide 24.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

Cf. A000142, A053150, A145642, A248762.

z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 3; Table[p[m, n], {n, 1, z}]  (* A248762 *)
Table[p[m, n]^(1/m), {n, 1, z}]   (* A248763 *)
Table[n!/p[m, n], {n, 1, z}]      (* A145642 *)
f[p_, e_] := p^Floor[e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 01 2024 *)

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(f[i, 2]\3));} \\ Amiram Eldar, Sep 01 2024

From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A053150(n!).
a(n) = (n! / A145642(n))^(1/3) = A248762(n)^(1/3).
log(a(n)) = (1/3)*n*log(n) - (log(3)+1)*n/3 + o(n) (Jakimczuk, 2017). (End)

A145643 Cubefree part of n!!.

Original entry on oeis.org

1, 2, 3, 1, 15, 6, 105, 6, 35, 60, 385, 90, 5005, 1260, 75075, 315, 1276275, 210, 24249225, 525, 509233725, 11550, 11712375675, 34650, 2342475135, 900900, 2342475135, 3153150, 67931778915, 28028, 2105885146365, 14014, 2573859623335

Offset: 1

Artur Jasinski, Oct 15 2008

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A004709, A050985, A006882, A069705 (cubefree part of 2^n), A145642.

CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n!! ], {n, 1, 40}]

a(n) = my(f = factor(prod(i = 0, (n-1)\2, n - 2*i))); prod(i = 1, #f~, f[i, 1]^(f[i, 2] % 3)); \\ Amiram Eldar, Sep 07 2024

from sympy import factorint, prod
def A145643(n):
    return 1 if n <= 1 else prod(p**(e % 3) for p, e in factorint(prod(range(n,0,-2))).items())
# Chai Wah Wu, Feb 04 2015

a(n) = A050985(A006882(n)). - Michel Marcus, May 11 2020

A248779 Rectangular array, by antidiagonals: T(m,n) = greatest (m+1)-th-power-free divisor of n!.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 6, 6, 2, 1, 30, 3, 6, 2, 1, 5, 15, 24, 6, 2, 1, 35, 90, 120, 24, 6, 2, 1, 70, 630, 45, 120, 24, 6, 2, 1, 70, 630, 315, 720, 120, 24, 6, 2, 1, 7, 210, 2520, 5040, 720, 120, 24, 6, 2, 1, 77, 2100, 280, 1260, 5040, 720, 120, 24, 6, 2, 1, 231

Offset: 1

Clark Kimberling, Oct 14 2014

Row 1:  A055204, greatest squarefree divisor of n!
Row 2:  A145642, greatest cubefree divisor of n!
Row 3:  A248766, greatest 4th-power-free divisor of n!
Rows 4 to 7:  A248769, A248772, A248775, A248778.
(The divisors are here called "greatest" rather than "largest" because the name refers to ">", called "greater than".)

			Northwest corner:
1   2   6   6   30   5    35    70
1   2   6   3   15   90   630   630
1   2   6   24  120  45   315   2520
1   2   6   24  120  720  5040  1260

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A055204, A145642, A248766, A248769, A248772, A248775, A248778, A000142.

f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}]
t = Table[n!/p[m, n], {m, 2, 16}, {n, 1, 16}]; TableForm[t]  (* A248779 array *)
f = Table[t[[n - k + 1, k]], {n, 12}, {k, n, 1, -1}] // Flatten (* A248779 seq. *)

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A248762 Greatest cube that divides n!.

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

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A248763 Greatest k such that k^3 divides n!

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A145643 Cubefree part of n!!.

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A248779 Rectangular array, by antidiagonals: T(m,n) = greatest (m+1)-th-power-free divisor of n!.

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A145644 Cubefree part of 10^n.

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