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%I #19 Jul 25 2021 01:30:42
%S 1,2,5,11,17,24,30,39,42,45,57,72,60,84,90,117,123,144,120,105,162,
%T 150,180,237,165,264,288,195,231,240,210,285,255,336,396,378,438,357,
%U 399,345,519,315,504,465,390,480,435,462,450,567,717,420,495,651,540,615,759,525,570,693,645
%N Least positive integer k such that 2k can be expressed as the sum of two primes in exactly n ways.
%C It appears that 2, 3, 4, 6 are the only numbers k such that 2k can be expressed as the sum of two primes in only one way.
%C Except when n = 1, a(n) = A258713(n). The first 11 terms of this sequence are the same as the initial terms of A053033. If a(n) exists for all n then A053033 is a subsequence. - _Andrew Howroyd_, Jan 28 2020
%H David A. Corneth, <a href="/A136244/b136244.txt">Table of n, a(n) for n = 0..16805</a> (first 1001 terms from Andrew Howroyd)
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F From _Andrew Howroyd_, Jan 28 2020: (Start)
%F a(n) = A023036(n) / 2.
%F A045917(a(n)) = n. (End)
%e a(3) = 11: 22 = 3 + 19 = 5 + 17 = 11 + 11. Also 22 is the least number which could be expressed as the sum of two prime numbers in exactly three ways.
%o (PARI) a(n, lim=oo)={for(i=1, lim, my(s=0); forprime(p=2, i, s+=isprime(2*i-p)); if(s==n, return(i))); -1} \\ _Andrew Howroyd_, Jan 28 2020
%Y Cf. A023036, A045917, A053033, A126204, A258713.
%K nonn
%O 0,2
%A K. B. Subramaniam (shunya_1950(AT)yahoo.co.in), Dec 24 2007
%E a(0)=1 prepended, a(5) corrected and a(7) and beyond from _Andrew Howroyd_, Jan 28 2020