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A134143 Let T(n) = (p, p+2) denote the n-th pair of twin primes. Let S(n) = 2p+2 (see A054735). Then a(n) = number of ways of writing S(n) as S(i) + S(j) with i <= j < m.

%I A134143 #20 Jul 02 2025 16:02:02
%S A134143 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,
%T A134143 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,
%U A134143 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
%N A134143 Let T(n) = (p, p+2) denote the n-th pair of twin primes. Let S(n) = 2p+2 (see A054735). Then a(n) = number of ways of writing S(n) as S(i) + S(j) with i <= j < m.
%C A134143 It is conjectured that a(n) > 0 for n >= 3.
%D A134143 R. K. Guy, ed., Unsolved Problems, Western Number Theory Meeting, Las Vegas, 1988.
%H A134143 Dmitry Kamenetsky, <a href="/A134143/b134143.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..680 from James A. Sellers and R. J. Mathar)
%e A134143 a(4) = 1 because S(4) = 17+19 = (5+7) + (11+13) = S(2)+S(3) and this is the only such way to write S(4) as the sum S(i) + S(j) for i <= j < 4.
%p A134143 with(numtheory): Sset := {}; for i from 1 to 5000 do if ithprime(i + 1) - ithprime(i) = 2 then Sset := Sset union {2 ithprime(i) + 2} fi; od; Sset := convert(Sset, list): for n from 1 to nops(Sset) do count := 0: s := Sset[n]: for i from 1 to n do if member(s - Sset[i], Sset) and s - Sset[i] >= s/2 then count:=count + 1 fi: od: printf(`%d,`, count): od:# _James Sellers_, Jan 25 2008
%p A134143 A134143 := proc(n)
%p A134143     local Sn, i, j, a;
%p A134143     Sn := A054735(n);
%p A134143     a := 0;
%p A134143     for i from 1 to n-1 do
%p A134143         for j from i to n-1 do
%p A134143         if A054735(i)+A054735(j) = Sn then
%p A134143             a := a+1;
%p A134143         end if;
%p A134143         end do:
%p A134143     end do:
%p A134143     a ;
%p A134143 end proc: # _R. J. Mathar_, Jan 25 2008
%Y A134143 Cf. A001359, A014574, A054735.
%K A134143 nonn
%O A134143 1,7
%A A134143 _N. J. A. Sloane_, Jan 25 2008
%E A134143 Terms a(5) onwards computed by _James Sellers_ and _R. J. Mathar_, Jan 25 2008