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A321498 Numbers which can be written in at least two ways in the form (x-y)*(x^2-y^2) with x > y > 0. 2

%I

%S 45,63,81,96,99,117,128,135,153,160,171,175,189,192,207,224,225,243,

%T 256,261,275,279,288,297,315,320,325,333,351,352,360,369,375,384,387,

%U 405,416,423,425,432,441,448,459,475,477,480,495,504,512,513,525,531,539,544,549,567,575,576,585

%N Numbers which can be written in at least two ways in the form (x-y)*(x^2-y^2) with x > y > 0.

%C 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.

%H David A. Corneth, <a href="/A321498/b321498.txt">Table of n, a(n) for n = 1..13937</a> (terms < 10^5)

%H Geoffrey B. Campbell, <a href="https://www.linkedin.com/groups/4510047/4510047-6434733867093057540">(m-n)(m^2-n^2) in two different ways</a>, LinkedIn Number Theory Group, Aug. 2018.

%e 45 = (4 - 1)*(4^2 - 1^2) = (23 - 22)*(23^2 - 22^2),

%e 63 = (5 - 2)*(5^2 - 2^2) = (32 - 31)*(32^2 - 31^2),

%e 81 = (6 - 3)*(6^2 - 3^2) = (41 - 40)*(41^2 - 40^2),

%e 96 = (5 - 1)*(5^2 - 1^2) = (13 - 11)*(13^2 - 11^2),

%e 99 = (7 - 4)*(7^2 - 4^2) = (50 - 49)*(50^2 - 49^2),

%e 117 = (8 - 5)*(8^2 - 5^2) = (59 - 58)*(59^2 - 58^2).

%t 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 *)

%o (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

%o (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

%Y Cf. A321499.

%K nonn

%O 1,1

%A Geoffrey B. Campbell and _M. F. Hasler_, Nov 22 2018

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Last modified August 3 08:21 EDT 2021. Contains 346435 sequences. (Running on oeis4.)