[seqfan] Re: I need a name for this sequence
Vladimir Shevelev
shevelev at bgu.ac.il
Fri Jun 10 15:45:26 CEST 2011
An infinite set of even N given by formula 8*u*v*(4*u*v^2+4*u*v+u-v) in view of the identity:
if x=4*u*v+2*u, y=8*u*v^2+4*u*v-2*v, z=4*u*v, then N=(x+y)z=xy+z.
Regards,
Vladimir
----- Original Message -----
From: Jose Miguel Ortiz Rodriguez <josemortiz at yahoo.com>
Date: Friday, June 10, 2011 9:26
Subject: [seqfan] Re: I need a name for this sequence
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> 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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Shevelev Vladimir
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