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A088915 Nonnegative numbers of the form mn(m+n) with integers m,n. 3

%I #29 Jul 03 2023 16:40:37

%S 0,2,6,12,16,20,30,42,48,54,56,70,72,84,90,96,110,120,126,128,132,156,

%T 160,162,180,182,198,210,240,250,264,272,286,306,308,324,330,336,342,

%U 380,384,390,420,432,448,462,468,506,510,520,540,546,552,560,576,600

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

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

%C 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 Contribution from _R. J. Mathar_, Oct 24 2010: (Start)

%C Examples of entries with more than one representation are:

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

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

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

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

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

%H T. D. Noe, <a href="/A088915/b088915.txt">Table of n, a(n) for n = 1..1000</a>

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

%p filter:= proc(n) local d,nd,x,y;

%p d:= numtheory:-divisors(n);

%p nd:= nops(d);

%p for x from 1 to nd do

%p for y from 1 to x do

%p if d[x]*d[y]*(d[x]+d[y])=n then return true fi

%p od od:

%p false

%p end proc:

%p filter(0):= 0:

%p select(filter, [seq(i,i=0..1000,2)]); # _Robert Israel_, Jul 24 2018

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

%o (Python)

%o from itertools import count, islice

%o from sympy import divisors, integer_nthroot

%o def A088915_gen(startvalue=0): # generator of terms >= startvalue

%o for m in count(max(startvalue,0)):

%o if m == 0:

%o yield m

%o else:

%o for k in divisors(m,generator=True):

%o p, q = integer_nthroot(k**4+(k*m<<2),2)

%o if q and not (p-k**2)%(k<<1):

%o yield m

%o break

%o A088915_list = list(islice(A088915_gen(),20)) # _Chai Wah Wu_, Jul 03 2023

%Y Cf. A121741.

%K nonn

%O 1,2

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

%E More terms from _Hugo Pfoertner_, Apr 10 2004

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)