A088915 Nonnegative numbers of the form mn(m+n) with integers m,n.

0, 2, 6, 12, 16, 20, 30, 42, 48, 54, 56, 70, 72, 84, 90, 96, 110, 120, 126, 128, 132, 156, 160, 162, 180, 182, 198, 210, 240, 250, 264, 272, 286, 306, 308, 324, 330, 336, 342, 380, 384, 390, 420, 432, 448, 462, 468, 506, 510, 520, 540, 546, 552, 560, 576, 600

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 29 2003

Keywords

• nonn

Comments

These are the values of 3 X 3 Vandermonde determinants with integer entries.

Solutions (m,n) are integral points on the elliptic curve m*n*(m+n)=a(n). Entries with record number of solutions are: 2, 6, 30, 240, 6480, 18480, 147840, 3991680 Possibly not minimal: a(n)=988159766157083520000000 has 22 solutions a(n)=2880932262848640000 20 solutions Multiplication of a(n) by u^3 does not decrease the number of solutions. [From Georgi Guninski, Oct 25 2010]

Contribution from R. J. Mathar, Oct 24 2010: (Start)

Examples of entries with more than one representation are:

- 30 = 5*1*6 = 3*2*5,

- 240 = 15*1*16 = 10*2*12 = 6*4*10, 6480 = 80*1*81 = 45*3*48 = 30*6*36 = 18*12*30,

- 18408 = 77*3*80 = 66*4*70 = 48*7*55 = 30*14*44 = 22*20*42,

- 147840 = 384*1*385 = 154*6*160 = 132*8*140 = 96*14*110 = 60*28*88 = 44*40*84 (6 representations),

- 110270160 = 6*4284*4290 = 60*1326*1386 = 66*1260*1326 = 102*990*1092 = ... with 8 representations. (End)

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A121741.

Programs

Maple

filter:= proc(n) local d,nd,x,y;
  d:= numtheory:-divisors(n);
  nd:= nops(d);
  for x from 1 to nd do
    for y from 1 to x do
      if d[x]*d[y]*(d[x]+d[y])=n then return true fi
  od od:
  false
end proc:
filter(0):= 0:
select(filter, [seq(i,i=0..1000,2)]); # Robert Israel, Jul 24 2018

Mathematica

Select[Range[0, 600], {} != FindInstance[m n (m + n) == # && n >= 0 && m >= 0, {m, n}, Integers, 1] &] (* Giovanni Resta, Jul 24 2018 *)

Python

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A088915_gen(startvalue=0): # generator of terms >= startvalue
    for m in count(max(startvalue,0)):
        if m == 0:
            yield m
        else:
            for k in divisors(m,generator=True):
                p, q = integer_nthroot(k**4+(k*m<<2),2)
                if q and not (p-k**2)%(k<<1):
                    yield m
                    break
A088915_list = list(islice(A088915_gen(),20)) # Chai Wah Wu, Jul 03 2023

Formula

a(n) = 2 * A121741(n-1) for n>=2.

Extensions

More terms from Hugo Pfoertner, Apr 10 2004