OFFSET
1,1
COMMENTS
An equivalent form is (x - y)^2*(x + y), or d^2*(d + 2y), where d = x - y > 0 and y > 0. See also A321499.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..13937 (terms < 10^5)
Geoffrey B. Campbell, (m-n)(m^2-n^2) in two different ways, LinkedIn Number Theory Group, Aug. 2018.
EXAMPLE
45 = (4 - 1)*(4^2 - 1^2) = (23 - 22)*(23^2 - 22^2),
63 = (5 - 2)*(5^2 - 2^2) = (32 - 31)*(32^2 - 31^2),
81 = (6 - 3)*(6^2 - 3^2) = (41 - 40)*(41^2 - 40^2),
96 = (5 - 1)*(5^2 - 1^2) = (13 - 11)*(13^2 - 11^2),
99 = (7 - 4)*(7^2 - 4^2) = (50 - 49)*(50^2 - 49^2),
117 = (8 - 5)*(8^2 - 5^2) = (59 - 58)*(59^2 - 58^2).
MATHEMATICA
aQ[n_] := Length[Solve[(x-y)*(x^2-y^2) ==n && x > y && y > 0, {x, y}, Integers]] > 1; Select[Range[600], aQ] (* Amiram Eldar, Dec 06 2018 *)
PROG
(PARI) select( is_A321498(n, c=2)={n&&!issquarefree(n)&&fordiv(n, d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d, 0)||c--||return(1))}, [0..999]) \\ Define the function is_A321498(). \\ ~30% speed up by David A. Corneth, Nov 23 2018
(PARI) is(n) = {if(issquarefree(n), return(0)); if(n % 2 == 0, if(n % 8 == 0, n\=8, return(0))); f = factor(n); e = select(x -> x > 1, f[, 2], 1); if(#e == 0 || n == 1, return(0), k = e[1]); n > f[k, 1]^3} \\ David A. Corneth, Dec 01 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey B. Campbell and M. F. Hasler, Nov 22 2018
STATUS
approved