cp's OEIS Frontend

A359078 a(n) is the first positive number that can be represented in exactly n ways as the sum of cubes of three distinct integers in arithmetic progression.

%I A359078 #13 Dec 31 2022 11:38:49
%S A359078 9,99,792,3829608,255816,24814152,198513216,1588105728,669982104,
%T A359078 5359856832,42878854656,7133969443392,57071755547136
%N A359078 a(n) is the first positive number that can be represented in exactly n ways as the sum of cubes of three distinct integers in arithmetic progression.
%C A359078 a(n) is the first positive number that can be represented in exactly n ways as 3*x*(x^2+2*y^2) = (x-y)^3 + x^3 + (x+y)^3 for positive numbers x and y.
%C A359078 Note that the first term x-y of the arithmetic progression need not be positive.
%C A359078 From _Ondrej Kutal_, Dec 21 2022: (Start)
%C A359078 a(14) <= 41605309793862144.
%C A359078 a(15) <= 35049191875384896000.
%C A359078 a(16) <= 25550860877155589184000. (End)
%e A359078 a(3) = 792 because 792 = (-6)^3 + 2^3 + 10^3 = (-1)^3 + 4^3 + 9^3 = 4^3 + 6^3 + 8^3 is the first number that can be represented in exactly 3 ways.
%p A359078 N:= 5*10^10:
%p A359078 L:= NULL:
%p A359078 for a from 1 while 3*a^3 <= N do
%p A359078   for b from 1 do
%p A359078     x:= 3*a*(a^2 + 2*b^2);
%p A359078     if x > N then break fi;
%p A359078     L:= L,x
%p A359078 od od:
%p A359078 L:= sort([L]):
%p A359078 V:= Vector(11):
%p A359078 m:= L[1]: count:= 1:
%p A359078 for i from 2 to nops(L) do
%p A359078 if L[i] = m then count:= count+1
%p A359078 else
%p A359078   if V[count] = 0 then V[count]:= m fi;
%p A359078   count:= 1; m:= L[i];
%p A359078 fi
%p A359078 od:
%p A359078 convert(V,list);
%Y A359078 Cf. A359030.
%K A359078 nonn,more
%O A359078 1,1
%A A359078 _Robert Israel_, Dec 15 2022
%E A359078 a(12)-a(13) from _Ondrej Kutal_, Dec 21 2022