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Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.
3

%I #14 Mar 16 2019 15:28:50

%S 0,1,1,2,1,2,2,2,2,2,2,2,3,3,1,3,3,3,3,4,3,4,6,3,2,4,3,4,5,3,2,5,4,4,

%T 5,4,4,7,4,4,5,3,6,7,3,5,7,4,4,7,4,5,10,5,4,7,3,7,9,5,6,8,5,5,9,5,5,

%U 11,6,5,9,5,6,10,5,6,8,6,6,9,5,5,12,6,5,9

%N Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

%C Conjecture. For n >= 1, a(n) > 0. This conjecture does not follow from the validity of the Goldbach binary conjecture because numbers of the form 4n+2, generally speaking, also have decompositions into sums of two primes of the form 4k+1.

%H Lei Zhou, <a href="/A156642/b156642.txt">Table of n, a(n) for n = 0..10000</a>

%e From _Lei Zhou_, Mar 19 2013: (Start)

%e n=1: 4n+2=6, 6=3+3; this is the only case that matches the definition, so a(1)=1;

%e n=3: 4n+2=14, 14=3+11=7+7; two instances found, so a(3)=2. (End)

%t Table[m = 4*n + 2; p1 = m + 1; ct = 0; While[p1 = p1 - 4; p2 = m - p1; p1 >= p2, If[PrimeQ[p1] && PrimeQ[p2], ct++]]; ct, {n, 1, 100}] (* _Lei Zhou_, Mar 19 2013 *)

%Y Cf. A002145, A002375, A061358, A002372, A045917, A214834, A217696.

%K nonn

%O 0,4

%A _Vladimir Shevelev_, Feb 12 2009