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%I #18 Sep 24 2018 16:53:14
%S 0,0,0,1,2,2,4,2,3,4,2,4,5,4,4,5,3,3,5,4,4,5,4,7,6,6,5,6,6,8,6,6,8,5,
%T 8,6,6,9,5,9,7,6,6,7,10,7,8,8,6,9,12,10,7,7,11,8,10,8,11,12,9,10,12,8,
%U 10,14,12,12,7,9,12,12,11,13,10,10,15,12,15,11,12,9,12,12,12,14,12,14,13
%N Number of ways of writing prime(n) in the form 2*prime(i)+prime(j).
%C First nonzero entry is for n=4: prime(4)=7=prime(1)+2*prime(3)=2+3*3, hence a(4)=1. Also, a(5)=2 because 11=5+2*3=7+2*2 (two solutions). Note that a(n) is not monotonic. - _Zak Seidov_, Jan 21 2006
%C Marnell conjectures that a(n) > 0 for n > 3. I find no exceptions below 10^9. - _Charles R Greathouse IV_, May 04 2010
%D Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.
%H Michael De Vlieger, <a href="/A103274/b103274.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A046926(prime(n)). - _David Wasserman_, Oct 08 2005
%e 11=2*2+7=2*3+5, so a(5)=2
%e a(100)=13 because p(100)=541=p(i)+2*p(j) for 13 pairs {i, j}: {2, 57}, {17, 53}, {23, 50}, {41, 42}, {49, 37}, {52, 36}, {56, 34}, {69, 25}, {76, 22}, {81, 18}, {91, 12}, {92, 11}, {96, 8}; e.g. 541=prime(96)+2*prime(8)=503+2*19. - _Zak Seidov_, Jan 21 2006
%t Table[Function[q, Length@ Select[#, Function[s, And[Length@ s == 2, Length@ First@ s == 1, MemberQ[Last@ , 2], Length@ Last@ s == 2]]] &@ Map[SortBy[Flatten[FactorInteger[#] /. {{p_, e_} /; e > 1 :> ConstantArray[p, e], {p_, 1} /; p > 1 :> p, {1, 1} -> 1}] & /@ #, Length] &, Select[IntegerPartitions[q, {2}], And[! MemberQ[#, 1], Total@ Boole@ PrimeQ@ # == 1] &]]]@ Prime@ n, {n, 89}] (* _Michael De Vlieger_, May 01 2017 *)
%o (PARI) a(n,q=prime(n))=my(s);forprime(p=2,q\2-1,if(isprime(q-2*p),s++));s \\ _Charles R Greathouse IV_, Jul 22 2015
%K nonn
%O 1,5
%A _Yasutoshi Kohmoto_, Jan 27 2005
%E More terms from _David Wasserman_, Oct 08 2005
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jul 14 2007
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