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 A141097 Number of unordered pairs of coprime composite numbers that sum to 2n. 6

%I #4 Mar 30 2012 17:22:50

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,1,0,0,2,0,1,3,0,1,

%T 0,0,3,3,0,0,3,0,3,2,0,3,4,0,1,2,1,4,5,0,0,3,1,5,5,0,4,6,0,5,2,0,7,6,

%U 0,0,9,2,8,8,0,6,4,1,8,4,1,9,9,1,4,10,2,8,11,0,3,11,3,10,4,2,12,8,1,4,13,2

%N Number of unordered pairs of coprime composite numbers that sum to 2n.

%C See A141095 for pairs of coprime nonprime numbers. It appears that a(n) > 0 except for the 43 values of 2n given in A141098. Roberts says that A. M. Vaidya proved that a(n) > 0 for all sufficiently large n.

%D Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 190.

%H T. D. Noe, <a href="/A141097/b141097.txt">Table of n, a(n) for n=1..10000</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_033.htm">Conjecture 33: The Goldbach Temptation</a>

%e a(17)=1 because 34 = 9+25.

%t Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

%Y Cf. A141099, A141100.

%K nonn

%O 1,29

%A _T. D. Noe_, Jun 02 2008

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