a new sequence

[David W. Wilson]

"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)