a new sequence

[Robert G. Wilson v]

Et al,

        I, earlier this morning wrote Neil to say that I have a problem with this sequence.  I now
suspect that the first entry after 1, 3, 5, 7, is over 10^5.  Sorry for the confusion.  Bob.

"David W. Wilson" wrote:

> "Robert G. Wilson v" wrote:
> >
> > Dear Neil,
> >
> >         What follows is my first attempt at the "proper" format.  Did
> > I get it right?  Bob.
> >
> >
> > > %I A000000
> > > %S A000000 1,3,5,7,2987,2989,3017,3019,3047,3049,3077,3079,3107,3109,3137,3139,3167,3169,
> > > %T A000000 3179,3181,3197,3199,3209,3211,3227,3229,3239,3241,3257,3259,3269,3271,3281,3283,
> > > %U A000000 3285,3287,3289,3299,3301,3311,3313,3315,3317,3319,3329,3331,3341,3343,3345,3347,3349,
> > > %N A000000 Odd numbers not the sum of three twin primes.
> > > %Y A000000 CF. A007534
> > > %R A000000 1,2
> > > %O A000000
> > > %K A000000 easy,
> > > %p A000000
> > > %t A000000 a=3;p=3;q=5;Do[r=Prime[n];If[p+2==q||q+2==r,a=Append[a,q]];p=q;q=r,{n,4,200}];
> > > %t A000000 l=Length[a];b=Union[Flatten[Table[a[[i]]+a[[j]]+a[[k]],{i,1,l},{j,1,l},{k,1,l}]]];
> > > %t A000000 c=Table[2i+1,{i,0,3a[[-1]]/2}];Complement[c,b]
> > > %A A000000 Robert G. Wilson v (rgwv at southwind.net)
> > > %H A000000
> >
>
> The format looks ok, but the sequence doesn't.  Here are reps
> of some of the larger sequence elements as twin + twin + twin:
>
> 2987 = 3 + 13 + 2971
> 2989 = 3 + 17 + 2969
> 3017 = 3 + 13 + 3001
> 3019 = 3 + 17 + 2999
> 3047 = 3 + 43 + 3001
> 3049 = 3 + 809 + 2237
> 3077 = 3 + 73 + 3001
> 3079 = 3 + 107 + 2969
> 3107 = 3 + 103 + 3001
> 3109 = 3 + 107 + 2999
> ...
>
> Among n <= 100000, my program finds only 1, 3, 5, 7 not of the
> form twin + twin + twin.  Indeed, most odd numbers appear to be of
> the form 3 + twin + twin.  It looks like the only exceptions are:
>
> 1 3 5 7 97 99 101 403 405 407 517 519 521 787 789 791 907 909 911 1117
> 1119 1121 1147 1149 1151 1267 1269 1271 1357 1359 1361 3247 3249 3251
> 4207 4209 4211 (no Sloane sequence)
>
> This is equivalent to the (old) conjecture that every even number
> is of the form twin + twin except
>
> 2 4 94 96 98 400 402 404 514 516 518 784 786 788 904 906 908 1114 1116
> 1118 1144 1146 1148 1264 1266 1268 1354 1356 1358 3244 3246 3248 4204
> 4206 4208 (A007534)