# 2 or 3 possibly new sequences from 1998

sellersj at math.psu.edu sellersj at math.psu.edu
Sat Jan 26 02:36:10 CET 2008

```Neil,

Just calculated the second sequence below for the first 5000 primes and
obtained the following:

5,8,12,18,52,100,946

So I only obtained one more term.

Take care.

James

>
> 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
>
>
>
>

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