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%I #19 Jul 04 2017 12:20:50
%S 0,3,45,204,1735,3640,13529,22271,58302,182507,250688,613046,1038538,
%T 1311167,2081983,3851401,6675916,7887193,12835857,17422352,20067757,
%U 30239982,39351402,56657043,88847952,110134459,121855019
%N Number of ways to represent prime(n)^3 as a sum of 3 distinct primes.
%C From the ternary Goldbach conjecture.
%H Charles R Greathouse IV, <a href="/A289193/b289193.txt">Table of n, a(n) for n = 1..50</a>
%F a(n) = A125688(A030078(n)).
%e a(2)=3: prime(2)=3, and 3^3=27=3+5+19=3+7+17=3+11+13 (3 ways).
%t Table[Length@ Select[IntegerPartitions[Prime[n]^3, {3}], And[AllTrue[#, PrimeQ], UnsameQ @@ #] &], {n, 7}] (* _Michael De Vlieger_, Jun 28 2017 *)
%o (PARI) a(n)=my(t=prime(n)^3,tp,s); forprime(p=t\3, t-4, tp=t-p; forprime(q=tp\2+1, min(tp, p-1), if(isprime(tp-q), s++))); s \\ _Charles R Greathouse IV_, Jul 02 2017
%Y Cf. A030078, A068307, A007963, A125688.
%K nonn
%O 1,2
%A _Zak Seidov_, Jun 28 2017
%E a(11)-a(27) from _Charles R Greathouse IV_, Jul 02 2017
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