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A337098 Least k whose set of divisors contains exactly n quadruples (x, y, z, w) such that x^3 + y^3 + z^3 = w^3, or 0 if no such k exists. 1
60, 120, 240, 432, 960, 360, 3840, 1728, 2592, 720, 1800, 2520, 161700, 1440, 6840, 9000, 2160, 2880, 168300, 5040, 41472, 5760, 1520820, 4320, 7200, 11520, 119700, 10080, 682080, 10800, 8640, 14400, 27360, 12960, 373248, 20160, 61560, 17280, 28800, 55440, 171000, 21600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Observation: a(n) == 0 (mod 12).

Listing primitive tuples (w, x, y, z) enables to compute for some m how many such tuples are in its divisors using the lcm of such tuples. - David A. Corneth, Sep 26 2020

REFERENCES

Y. Perelman, Solutions to x^3 + y^3 + z^3 = u^3, Mathematics can be Fun, pp. 316-9 Mir Moscow 1985.

LINKS

David A. Corneth, Table of n, a(n) for n = 1..504

Fred Richman, Sums of Three Cubes

EXAMPLE

a(3) = 240 because the set of the divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 quadruples {3, 4, 5, 6}, {6, 8, 10, 12} and {12, 16, 20, 24}. The first quadruple is primitive.

MAPLE

with(numtheory):divisors(240);

for n from 1 to 52 do :

ii:=0:

for q from 6 by 6 to 10^8 while(ii=0) do:

   d:=divisors(q):n0:=nops(d):it:=0:

    for i from 1 to n0-3 do:

     for j from i+1 to n0-2 do :

      for k from j+1 to n0-1 do:

      for m from k+1 to n0 do:

       if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3

        then

        it:=it+1:

        else

       fi:

      od:

     od:

    od:

    od:

    if it = n

     then

     ii:=1: printf (`%d %d \n`, n, q):

     else

    fi:

od:

od:

MATHEMATICA

With[{s = Array[Count[Subsets[Divisors[#], {4}]^3, _?(#1 + #2 + #3 == #4 & @@ # &)] &, 10^4]}, Rest@ Values[#][[1 ;; 1 + LengthWhile[Differences@ Keys@ #, # == 1 &] ]] &@ KeySort@ PositionIndex[s][[All, 1]]] (* Michael De Vlieger, Sep 18 2020 *)

PROG

(Python)

from itertools import combinations

from sympy import divisors

def A337098(n):

    k = 1

    while True:

        if n == sum(1 for x in combinations((d**3 for d in divisors(k)), 4) if sum(x[:-1]) == x[-1]):

            return k

        k += 1 # Chai Wah Wu, Sep 25 2020

CROSSREFS

Cf. A027750, A095868, A095867, A096545, A096546, A328204, A328149, A331365.

Sequence in context: A309842 A177871 A334382 * A252953 A309315 A275339

Adjacent sequences:  A337095 A337096 A337097 * A337099 A337100 A337101

KEYWORD

nonn,hard

AUTHOR

Michel Lagneau, Aug 15 2020

EXTENSIONS

a(13)-a(22) from Chai Wah Wu, Sep 25 2020

More terms from David A. Corneth, Sep 26 2020

STATUS

approved

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Last modified April 20 20:26 EDT 2021. Contains 343137 sequences. (Running on oeis4.)