%I
%S 1,4,8,12,18,24,30,39,42,45,57,72,60,84,90,117,123,144,120,105,162,
%T 150,180,237,165,264,288,195,231,240,210,285,255,336,396,378,438,357,
%U 399,345,519,315,504,465,390,480,435,462,450,567,717,420,495,651,540,615
%N Least number m such that 2m can be expressed as the sum of two distinct primes in exactly n ways.
%C Let f(x) = A061357(x) be the number of primes p < x such that 2xp is also prime. a(n) is the smallest positive integer x such that f(x) = n.
%C Or, least number m such that m can be expressed as the mean of two distinct primes in exactly n ways. Cf. A061357 = number of ways n can be expressed as the mean of two distinct primes, A061357 = number of ways the even integer 2n can be written as the sum of two primes for all even integers >6.  _Zak Seidov_, Sep 08 2006
%C For what values of n is a(n) > a(n+1)?
%H Reinhard Zumkeller, <a href="/A072511/b072511.txt">Table of n, a(n) for n = 0..1000</a>
%F It seems that for n>7 n*log(n)*log(log(n)) < a(n) < 3n*log(n)*log(log(n)). Does lim n>infinity a(n)/n/log(n)/log(log(n)) exist ?  _Benoit Cloitre_, Aug 11 2002
%e a(1)=4 because 8 = 3+5 that is 8 can be expressed as the sum of two distinct primes by exactly 1 way,
%e a(2)=8 because 16 = 3+13 = 5+11 (2 ways),
%e a(3)=12 because 24 = 5+17 = 7+17 = 11+17 (3 ways),
%e a(4)=18 because 36 = 5+31 = 7+29 = 13+23 = 17+19 (4 ways), etc.
%e Starting with third term 12, all terms are multiples of 3.
%t f[x_] := Length[Select[2x(Prime/@Range[PrimePi[x1]]), PrimeQ]]; For[x=1, x<1000, x++, fx=f[x]; If[a[fx]>=0, Null, Null, a[fx]=x]]; a/@Range[0, 60]
%o (Haskell)
%o import Data.List (elemIndex)
%o import Data.Maybe (fromJust)
%o a072511 = (+ 1) . fromJust . (`elemIndex` a061357_list)
%o  _Reinhard Zumkeller_, Nov 10 2012
%Y Cf. A061357, A061357.
%K nonn
%O 0,2
%A _Amarnath Murthy_, Jul 24 2002
%E Edited by _Dean Hickerson_, Aug 07 2002
%E Entry revised by _N. J. A. Sloane_, Sep 12 2006
