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

Jose Miguel Ortiz Rodriguez josemortiz at yahoo.com
Fri Jun 10 02:04:36 CEST 2011

```Looks similar to the precedence rules for boolean numbers:

Boolean Precedence:
AB = A·B
A·B+C = (A·B) + C
A+B·C = A + (B·C)

Your series seems to treat the numbers in a similar way to the boolean definition of De Morgan's Theorem, in the sense of  (A*B+C)'=(A'+B')*C' {and then both =N in your case} or something like that.

It's been a long time since my college years and my digital circuits theory courses, so this example that I wrote may be wrong. But I do remember something about flipping the *'s by the +'s directly when boolean negation was in the equation in order to keep the equality in the logic circuit while using different gates for the implementation. This would be like your equivalent 'digital' output N in this case.

Obviously you are not working just with 0's and 1's in your algorithm, but maybe looking at it from the variables point of view, there could be some relationship, at least to help you come up with a name for it. Hope it helps :-)

--- On Thu, 6/9/11, David Wilson <davidwwilson at comcast.net> wrote:

From: David Wilson <davidwwilson at comcast.net>
Subject: [seqfan] Re: I need a name for this sequence
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Date: Thursday, June 9, 2011, 6:38 PM

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

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