cp's OEIS Frontend

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

%I A088915 #29 Jul 03 2023 16:40:37
%S A088915 0,2,6,12,16,20,30,42,48,54,56,70,72,84,90,96,110,120,126,128,132,156,
%T A088915 160,162,180,182,198,210,240,250,264,272,286,306,308,324,330,336,342,
%U A088915 380,384,390,420,432,448,462,468,506,510,520,540,546,552,560,576,600
%N A088915 Nonnegative numbers of the form mn(m+n) with integers m,n.
%C A088915 These are the values of 3 X 3 Vandermonde determinants with integer entries.
%C A088915 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]
%C A088915 Contribution from _R. J. Mathar_, Oct 24 2010: (Start)
%C A088915 Examples of entries with more than one representation are:
%C A088915 - 30 = 5*1*6 = 3*2*5,
%C A088915 - 240 = 15*1*16 = 10*2*12 = 6*4*10, 6480 = 80*1*81 = 45*3*48 = 30*6*36 = 18*12*30,
%C A088915 - 18408 = 77*3*80 = 66*4*70 = 48*7*55 = 30*14*44 = 22*20*42,
%C A088915 - 147840 = 384*1*385 = 154*6*160 = 132*8*140 = 96*14*110 = 60*28*88 = 44*40*84 (6 representations),
%C A088915 - 110270160 = 6*4284*4290 = 60*1326*1386 = 66*1260*1326 = 102*990*1092 = ... with 8 representations. (End)
%H A088915 T. D. Noe, <a href="/A088915/b088915.txt">Table of n, a(n) for n = 1..1000</a>
%F A088915 a(n) = 2 * A121741(n-1) for n>=2.
%p A088915 filter:= proc(n) local d,nd,x,y;
%p A088915   d:= numtheory:-divisors(n);
%p A088915   nd:= nops(d);
%p A088915   for x from 1 to nd do
%p A088915     for y from 1 to x do
%p A088915       if d[x]*d[y]*(d[x]+d[y])=n then return true fi
%p A088915   od od:
%p A088915   false
%p A088915 end proc:
%p A088915 filter(0):= 0:
%p A088915 select(filter, [seq(i,i=0..1000,2)]); # _Robert Israel_, Jul 24 2018
%t A088915 Select[Range[0, 600], {} != FindInstance[m n (m + n) == # && n >= 0 && m >= 0, {m, n}, Integers, 1] &] (* _Giovanni Resta_, Jul 24 2018 *)
%o A088915 (Python)
%o A088915 from itertools import count, islice
%o A088915 from sympy import divisors, integer_nthroot
%o A088915 def A088915_gen(startvalue=0): # generator of terms >= startvalue
%o A088915     for m in count(max(startvalue,0)):
%o A088915         if m == 0:
%o A088915             yield m
%o A088915         else:
%o A088915             for k in divisors(m,generator=True):
%o A088915                 p, q = integer_nthroot(k**4+(k*m<<2),2)
%o A088915                 if q and not (p-k**2)%(k<<1):
%o A088915                     yield m
%o A088915                     break
%o A088915 A088915_list = list(islice(A088915_gen(),20)) # _Chai Wah Wu_, Jul 03 2023
%Y A088915 Cf. A121741.
%K A088915 nonn
%O A088915 1,2
%A A088915 Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 29 2003
%E A088915 More terms from _Hugo Pfoertner_, Apr 10 2004