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A306213 Numbers that are the sum of cubes of three distinct positive integers in arithmetic progression.

%I A306213 #24 Dec 17 2022 08:26:07
%S A306213 36,99,153,216,288,405,408,495,645,684,792,855,972,1071,1197,1224,
%T A306213 1407,1548,1584,1701,1728,1968,2079,2241,2304,2403,2541,2673,2736,
%U A306213 3051,3060,3240,3264,3537,3540,3888,3960,4059,4131,4257,4500,4587,4833,5049,5160,5256,5472,5643,5832,5940,6336,6369,6669
%N A306213 Numbers that are the sum of cubes of three distinct positive integers in arithmetic progression.
%C A306213 Numbers that can be written as 3*a*(a^2 + 2*b^2) = (a-b)^3 + a^3 + (a+b)^3 where 0 < b < a. - _Robert Israel_, Dec 15 2022
%H A306213 Robert Israel, <a href="/A306213/b306213.txt">Table of n, a(n) for n = 1..10000</a>
%e A306213 153 = 1^3 + 3^3 + 5^3, with 3 - 1 = 5 - 3 = 2;
%e A306213 972 = 3^3 + 6^3 + 9^3, with 6 - 3 = 9 - 6 = 3.
%p A306213 N:= 10000: # for terms <= N
%p A306213 S:= {}:
%p A306213 for a from 1 while a^3 + (a+1)^3 + (a+2)^3 <= N do
%p A306213   for d from 1 do
%p A306213     x:= a^3 + (a+d)^3 + (a+2*d)^3;
%p A306213     if x > N then break fi;
%p A306213     S:= S union {x}
%p A306213 od od:
%p A306213 sort(convert(S,list)); # _Robert Israel_, Dec 14 2022
%o A306213 (PARI) for(n=3, 7000, k=(n/3)^(1/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^3)/(6*a); if(b==truncate(b)&&issquare(b), d=sqrt(b), d=0); if(d>=1&&d<=a-1, v=1; print1(n,", ")); a+=1))
%o A306213 (PARI) w=List(); for(n=3, 7000, k=(n/3)^(1/3); for(a=2, k, for(c=1, a-1, v=(a-c)^3+a^3+(a+c)^3; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))
%Y A306213 Cf. A000578, A023042, A024975, A122723, A306212, A306214.
%K A306213 nonn
%O A306213 1,1
%A A306213 _Antonio Roldán_, Jan 29 2019