OFFSET
1,2
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
FORMULA
a(n) = 2 * A121741(n-1) for n>=2.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 29 2003
EXTENSIONS
More terms from Hugo Pfoertner, Apr 10 2004
STATUS
approved