2 or 3 possibly new sequences from 1998

[sellersj at math.psu.edu]

Neil,

Greetings!  I wrote up some quick Maple code to generate the first
sequence you mentioned below.  The values I computed appear below.

I would be happy to submit the sequence to the OEIS.  Can someone confirm
these values before I do so?  I would be most appreciative.

All the best.

James

0,0,1,1,1,1,2,2,2,1,1,2,3,2,3,1,4,3,3,3,2,6,3,5,3,3,3,3,3,8,4,2,3,3,6,4,4,6,7,8,3,6,3,9,8,7,7,5,8,4,1,6,6,3,7,1,6,6,4,8,1,5,5,8,9,11,10,6,8,16,13,9,12,6,7,8,4,16,9,6,13,10,9,5,6,6,8,11,16,11,13,6,6,6,17,9,6,6,4,14,12,6,10,12,13,10,9,8,12,12,13,12,11,16,9,17,13,4,8,6,17,8,18,13,8,10,8,7,13,7,22,9,9,9,12,16,16,7,14,11,26,10,21,21,12,6,23,19,20,12,17,13,15,16,10,15,12,18,5,13,9,18,13,14,15,10,24,9,16,16,19,31,14,12,15,25,10,17,18,10,11,18,17,9,10,25,13,14,17,24,12,19,28,21,12,22,15,23,21,18,12,16,8,19,13,13,19,14,17,21,12,18,19,21,16,24,25,26,14,29,10,18,16,20,24,16,21,18,37,18,21,16,15,23,34,13,15,16,11,18,16,19,21,17,18,6,19,14,12,13,30,23,18,18,15,16,37,10,14,10,12,20,16,31,20,11,21,16,24,11,18,16,13,20,20,27,23,18,19,12,16,16,17,18,10,14,20,16,14,13,14,25,17,17,29,13,25,16,23,19,26,19,22,25,23,23,16,25,9,19,21,22,27,10,24,12,17,41,30,20,19,11,21,24,22,20,15,17,38,25,16,27,20,16,20,14,24,18,18,31,18,33,23,23,26,18,19,10,16,30,20,20,39,20,16,13,18,24,18,22,11,23,25,20,16,22,2!
 1,34,15,20,21,54,23,18,37,15,21,14,8,15,35,22,30,31,22,22,35,21,16,27,26,16,18,20,16,41,18,30,36,18,14,28,37,59,19,30,18,23,18,43,16,19,47,26,28,23,35,15,16,33,28,33,29,19,36,25,31,22,38,24,19,40,19,19,25,15,39,17,38,26,46,18,36,23,30,24,27,38,41,21,31,47,24,32,25,28,24,20,20,24,17,25,45,32,37,24,28,16,31,20,40,20,14,30,12,34,17,38,48,36,37,21,27,43,26,31,32,37,10,34,22,16,23,29,27,31,40,16,45,24,15,15,30,35,29,38,24,34,35,48,24,53,55,19,40,34,20,45,20,34,29,61,27,35,12,18,23,34,20,26,34,35,33,27,56,20,24,36,21,21,30,58,29,25,47,19,47,29,33,24,19,32,32,22,32,41,35,25,42,31,57,26,32,22,37,32,56,16,17,39,25,22,12,18,35,32,63,29,47,28,15,33,17,18,28,43,23,21,52,22,27,24,29,44,15,52,28,48,29,24,21,28,24,15,30,25,46,41,32,34,22,22,31,45,33,34,28,39,49,33,40,63,50,52,38,33,25,24,27,45,20,27,85,31,25,40,53,28,47,36,40,28,20,41,56,59,31,48,50,27,23,33,34,26,16,31,29,31,52,50,39,38,35,60,43,23,20,41,34,35,63,37,42,19,34,41,34,16,47,59,



>
> Dear Seqfans,   I just came across a draft of a list
> of open problems from the Western Number Theory Meeting, Asilomar, 1988.
>
> This suggested two possibly new sequences - maybe
> someone would like to look into them?
>
> 1.
> Let T(n) := (p, p+2) denote the n-th pair of twin primes.
> Let S(n) = 2p+2.
>
> Then a(n) = number of ways of writing S(n) as S(i) + S(j) with i <= j < m.
> Sequence begins 0,0,1,1,...
>
> a(4) = 1 because s(4) = 17+19 = (5+7) + (11+13) = S(2)+S(3).
>
>
> 2.
>
> Consider A001043:
> %I A001043 M3780 N0968
> %S A001043
> 5,8,12,18,24,30,36,42,52,60,68,78,84,90,100,112,120,128,138,144,152,
> %T A001043
> 162,172,186,198,204,210,216,222,240,258,268,276,288,300,308,320,330,
> %U A001043
> 340,352,360,372,384,390,396,410,434,450,456,462,472,480,492,508,520
> %N A001043 Numbers that are the sum of 2 successive primes.
>
> The new sequence would be:
> Let m = A001043(n). Then a(n) = number of ways of writing
> m = A001043(i) + A001043(j) with i <= j < n.
>
> This is not always possible, and we have:
>
> %I A134650
> %S A134650 5,8,12,18,52,100
> %N A134650 Numbers n such that n is the sum of two consecutive primes but
> is not the sum of tw
> o sums of consecutive primes.
> %C A134650 Numbers in A001043 but not in A134651.
> %C A134650 Conjectured to be finite, may be complete.
> %O A134650 1,1
> %D A134650 R. K. Guy, ed., Unsolved Problems, Western Number Theory
> Meeting, Las Vegas, 1988.
> %K A134650 nonn,fini,more
> %A A134650 njas, Jan 25 2008
>
>
> Could someone work out these sequences?
>
> Neil
>
>
>
>