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

Vladimir Shevelev shevelev at bgu.ac.il
Sat Jun 11 19:11:24 CEST 2011


The set of all terms of the sequence is the set of different numbers given by the following two formulas with positive u,v:
1) a=u*v+u, b=u*v^2+u*v-v, c=u*v, if v>1, and
 a=2*u-1, b=2*u, c=u, if v=1.
2) a=u*v+u+1, b=u*v^2+u*v+v+1, c=u*v+1,
such that in all cases we have (a+b)*c=a*b+c=N.
Proof. Consider Diophantine equation (a+b)*c=a*b+c with the condition 0<a<b. Then c>0.
Put x-y=a, x+y=b. Since 2*x=a+b, then in order to not lose a solution, we suppose that x,y are integers or half-integers. Then we have 2*x*c=x^2-y^2+c, or (x-c)^2-y^2=c*(c-1). This equation splits into two disjoint cases: 1) x-c-y=c/v, x-c+y=v*(c-1), where v is a divisor of c;  2) x-c-y=(c-1)/v, x-c+y=v*c, where v is a divisor of c-1. Both cases are considered quite analogously. Therefore, let us consider 1). Put c=v*u. Then a=x-y=v*u+u, b=x+y=v*u+v^2*u-v and  1) follows.
Note that the case v=1 is only  case when a=2*u>b=2*u-1. Since the equation is (a,b)-symmetric, then  in this case we put a=2*u-1,b=2*u with c=u.
 <>
Note that full description of odd terms of the sequence are given by these formulas, if in 1) u,v are odd and in 2) u,v are of different parities.
 
Regards.
Vladimir


----- Original Message -----
From: Maximilian Hasler <maximilian.hasler at gmail.com>
Date: Friday, June 10, 2011 18:59
Subject: [seqfan] Re: I need a name for this sequence
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> apologies for re-posting, but
> 
> is_abc(N)={fordiv(N,c,c*c>N&return;c>1&fordiv(N-c,a,(a+(N-
> c)/a)*c==N&return(1)))}
> is over 250 times faster, for N=1..9999
> The "a>(N-c)/a & break" gave a significant speedup only in the 
> other version.
> 
> One can also prove that always a>c.
> (Assume a<=c, then N = ab+c <= cb+c < c(b+a) = N, 
> impossible.)(But adding this test in the inner loop seems not to 
> give a
> significant speedup.)
> 
> Maximilian
> 
> 
> On Fri, Jun 10, 2011 at 11:09 AM, Maximilian Hasler
> <maximilian.hasler at gmail.com> wrote:
> > I agree with David on the missing requirement
> > to exclude the "trivial" decomposition (1,N-1,1) for any N.
> > (For example, to require c>1 is enough).
> >
> > So the definition of the sequence could be :
> >
> > Numbers N such that N=(a+b)*c=a*b+c for some a,b,c>1.
> >
> > and a more "playful" characterization would be to speak about 
> exchange> of + and * in the expression (a+b)*c.
> >
> > PARI code:
> >
> > is_abc(N)={for(c=2,N-2,fordiv(N-c,a,a>(N-c)/a&break;(a+(N-
> c)/a)*c==N&return(1)))}>
> > for(N=1,1999,is_abc(N)&print1(N","))
> >
> > Maximilian
> >
> >
> >
> >
> > On Fri, Jun 10, 2011 at 9:45 AM, Vladimir Shevelev 
> <shevelev at bgu.ac.il> wrote:
> >>
> >> 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
> >>>
> >>>
> >>> _______________________________________________
> >>>
> >>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>
> >>> _______________________________________________
> >>>
> >>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>
> >>
> >>  Shevelev Vladimir‎
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 

 Shevelev Vladimir‎



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