[seqfan] Re: I need a name for this sequence

[David Wilson]

If you allow a, b, c nonnegative, then (a, b, c) = (0, 1, N) implies 
every N >= 0 is one of your numbers.

If you require a, b, c positive, then (a, b, c) = (1, N-1, 1) implies 
every N >= 2 is one of your numbers.

Perhaps you meant to require a, b, c >= 2. This seems to generate the 
numbers you seek.

I'm supposing this is the correct constraint, and that your list below 
is meant to list all elements up
to 10000. In that case, your list is incomplete. The first missing 
element I found was:

     1694 = (16+105)*14 = (16*105)+14

My Perl program below:

my $N = 10000;
my %seen = ();
for (my $b = 2; 2*($b+1) <= $N; $b++) {
print "b = $b\n";
     my $v = $b*($b-1);
     for (my $d = $b+1; $d <= $v; $d++) {
         next if $v%$d;
         my $a = ($d-$b+1);
         last if $a > $b;
         my $c = ($a*$b)/$d;
         my $k = $a*$b+$c;
         next if $k > $N;
         $seen{$k} = 1;
     }
}
my @k = sort {$a <=> $b} keys %seen;
print map("$_\n", @k);

generates the hopefully complete list

14 33 39 60 64 84 95 110 138 150 155 174 189 217 248 258 259 272 315 324 
360 368
  390 399 405 410 430 473 504 530 539 564 584 624 663 670 732 754 770 
819 852 854
  869 885 897 915 1005 1008 1024 1053 1056 1065 1104 1110 1120 1139 1155 
1248 1278
1292 1360 1378 1422 1425 1463 1536 1545 1580 1615 1674 1694 1743 1760 
1785 1802
1806 1840 1869 1884 1914 1919 1974 2002 2055 2093 2134 2280 2289 2369 
2379 2420
2464 2475 2478 2500 2538 2544 2574 2625 2678 2751 2780 2794 2800 2889 
2924 2945
2954 2990 2997 3000 3108 3164 3171 3248 3267 3302 3325 3335 3438 3472 
3504 3570
3615 3668 3770 3807 3813 3885 3900 3990 4009 4064 4172 4309 4323 4368 
4375 4422
4488 4544 4560 4590 4640 4710 4779 4788 4794 4862 4865 4884 5100 5115 
5134 5148
5184 5219 5220 5249 5264 5334 5439 5478 5495 5508 5580 5640 5709 5738 
5760 5915
6045 6094 6142 6154 6156 6174 6272 6279 6325 6330 6336 6360 6422 6480 
6640 6683
6760 6798 6804 6825 6913 6923 6965 6972 7014 7120 7170 7239 7289 7353 
7488 7544
7685 7700 7824 7843 7854 7857 7956 7960 7995 8055 8145 8184 8235 8299 
8370 8418
8420 8520 8547 8624 8645 8708 8786 8789 8800 8874 8880 9090 9168 9282 
9324 9405
9430 9555 9604 9723 9810 9840 9847 9854 9917 9950

On 6/9/2011 4:39 PM, Claudio Meller wrote:
> Take a,b and c with a<b and a<>b,  b<>c
>
> I search for numbers N such N = (a+b) x c = (axb) + c
>
> For example :
> a    b   c     N
> (3 , 4 , 2) = 14
> (5 , 6 , 3) = 33
> (4 , 9 , 3) = 39
>
>
> Sequence :
> 14, 33, 39, 60, 64, 84, 95, 110, 138, 150, 155, 174, 189, 217, 248, 258,
> 259, 272, 315, 324, 360, 368, 390, 399, 405, 410, 430, 473, 504, 530, 539,
> 564, 584, 624, 663, 670, 732, 754, 770, 819, 852, 854, 869, 885, 897, 915,
> 1005, 1008, 1024, 1053, 1056, 1065, 1104, 1110, 1120, 1139, 1155, 1248,
> 1278, 1292, 1360, 1378, 1422, 1425, 1536, 1545, 1580, 1615, 1674, 1743,
> 1760, 1785, 1802, 1806, 1840, 1869, 1914, 1919, 1974, 2002, 2093, 2134,
> 2280, 2289, 2369, 2420, 2475, 2478, 2500, 2544, 2574, 2625, 2678, 2794,
> 2800, 2889, 2990, 2997, 3000, 3000, 3108, 3164, 3248, 3267, 3302, 3325,
> 3335, 3472, 3570, 3668, 3770, 3813, 3990, 4064, 4309, 4323, 4422, 4488,
> 4544, 4590, 4794, 4865, 4884, 5134, 5148, 5184, 5220, 5439, 5508, 5580,
> 5738, 6045, 6279, 6360, 6480, 6640, 6683, 7014, 7353, 7700, 8055, 8145,
> 8418, 8789, 9168, 9555, 9950
>
> How can I define this sequence?
>
> Thanks