

A328149


Numbers whose set of divisors contains a quadruple (x, y, z, w) satisfying x^3 + y^3 + z^3 = w^3.


4



60, 72, 120, 144, 180, 216, 240, 288, 300, 360, 420, 432, 480, 504, 540, 576, 600, 648, 660, 720, 780, 792, 840, 864, 900, 936, 960, 1008, 1020, 1080, 1140, 1152, 1200, 1224, 1260, 1296, 1320, 1368, 1380, 1440, 1500, 1512, 1560, 1584, 1620, 1656, 1680, 1710
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OFFSET

1,1


COMMENTS

The subsequence of numbers of the form 2^i*3^j is 72, 144, 216, 288, 432, 576, 648, 864, 1152, 1296, ...
The corresponding number of quadruples of the sequence is 1, 1, 2, 2, 2, 2, 3, 3, 2, 6, 2, 4, 4, 2, 3, 4, 4, 3, 2, 10, ... (see the sequence A328204).
The set of divisors of a(n) contains at least one primitive quadruple.
Examples:
The set of divisors of a(1) = 60 contains only one primitive quadruple: (3, 4, 5, 6).
The set of divisors of a(10) = 360 contains two primitive quadruples: (1, 6, 8, 9) and (3, 4, 5, 6).
From Robert Israel, Jul 06 2020: (Start)
Every multiple of a member of the sequence is in the sequence.
The first member of the sequence not divisible by 6 is a(68) = 2380, which has the quadruple (7, 14, 17, 20).
The first odd member of the sequence is a(1230) = 43065, which has the quadruple (11, 15, 27, 29). (End)


REFERENCES

Y. Perelman, Solutions to x^3 + y^3 + z^3 = u^3, Mathematics can be Fun, pp. 3169 Mir Moscow 1985.


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000
Fred Richman, Sums of Three Cubes


EXAMPLE

120 is in the sequence because the set of divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120} contains the quadruples {3, 4, 5, 6} and {6, 8, 10, 12}. The first quadruple is primitive.


MAPLE

with(numtheory):
for n from 3 to 2000 do :
d:=divisors(n):n0:=nops(d):it:=0:
for i from 1 to n03 do:
for j from i+1 to n02 do :
for k from j+1 to n01 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>0 then
printf(`%d, `, n):
else fi:
od:


MATHEMATICA

nq[n_] := If[ Mod[n, 6]>0, 0, Block[{t, u, v, c = 0, d = Divisors[n], m}, m = Length@ d; Do[ t = d[[i]]^3 + d[[j]]^3; Do[u = t + d[[h]]^3; If[u > n^3, Break[]]; If[ Mod[n^3, u] == 0 && IntegerQ[v = u^(1/3)] && Mod[n, v] == 0, c++], {h, j+1, m  1}], {i, m3}, {j, i+1, m  2}]; c]]; Select[ Range@ 1026, nq[#] > 0 &] (* program from Giovanni Resta adapted for the sequence. See A330893 *)


PROG

(PARI) isok(n) = {my(d=divisors(n), m); if (#d > 3, for (i=1, #d3, for (j=i+1, #d2, for (k=j+1, #d1, if (ispower(d[i]^3+d[j]^3+d[k]^3, 3, &m) && !(n%m), return (1)); ); ); ); ); } \\ Michel Marcus, Nov 15 2020


CROSSREFS

Cf. A023042, A027750, A096545, A328204, A330893.
Sequence in context: A347935 A347938 A114837 * A154548 A244384 A290018
Adjacent sequences: A328146 A328147 A328148 * A328150 A328151 A328152


KEYWORD

nonn


AUTHOR

Michel Lagneau, Jun 07 2020


STATUS

approved



